December 19, 2008

Buckingham gets hot in cool venue

Buckingham Charter Magnet High School's boys basketball team found a way to stay hot Monday, even without heat.

James Alexander scored 17 points as the Knights downed North Hills Christian of Vallejo 47-32 at the cavernous and chilly Mare Island Sports Complex.

"It must have been 50 degrees in there. My assistant and I never got our jackets off," said Dan Curry, Buckingham's head coach. "I didn't see too many of the kids sweating. I just told them you have to learn how to play with adversity, and this was just another kind of adversity. I was really proud of them."

Marcus Rojas added 10 points and 10 rebounds, his fourth straight double-double, as the Knights improved to 4-6 overall.

Buckingham led throughout, but North Hills cut the lead to eight points in the third quarter before Matt Poirier hit a 3-pointer. Poirier finished with nine points and five assists.

The two teams play again Saturday at noon at Vaca Pena Middle School.


Source: The Reporter

Mother of Bristol Palin's fiance arrested

http://www.reuters.com/resources/r/?m=02&d=20081219&t=2&i=7283569&w=450&r=2008-12-19T204733Z_01_BTRE4BI1LNO00_RTROPTP_0_USA-POLITICS

The mother of an 18-year-old man who plans to marry Alaska Gov. Sarah Palin's pregnant daughter, Bristol, has been arrested on drug charges, authorities said on Friday.

Sherry Johnston, 42, was taken into custody at her home in Wasilla, Alaska, on Thursday after an undercover narcotics investigation, Alaska State Troopers said in a statement.

She faces six counts of misconduct involving a controlled substance, said trooper spokeswoman Megan Peters. She was released on $5,000 bail shortly after her arrest. All the charges are felonies.

Peters declined to release any further information about the case, saying police needed to protect the investigation.

Sarah Palin, the Republican vice presidential nominee in the 2008 election, revealed during the campaign her 18-year-old daughter was pregnant and that Levi Johnston was the father.

The two have said they plan to marry and Bristol Palin is due to give birth this weekend.


Source: Reuters

Best pictures of Fairies and Fairy Tales

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Disney Interactive Studios Releases Disney Fairies: Tinker Bell for Nintendo DS

Disney Interactive Studios today announced the release of Disney Fairies: Tinker Bell for Nintendo DS(TM). The game expands on Tinker Bell's adventures in the all-new movie "Tinker Bell" from Walt Disney Studios.


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Home Entertainment available now on Disney DVD and Blu-ray(R) Hi-Def and connects with the recently launched Disney Fairies Pixie Hollow (www.PixieHollow.com) virtual world from Disney Online. Players will be able to see environments and characters that are featured in the movie while playing the game, and also use codes found in-game to unlock items in the virtual world.

"Disney Fairies: Tinker Bell introduces girls to countless hours of original gameplay that is unmatched in quality and design," said Craig Relyea, senior vice president of global marketing, Disney Interactive Studios. "Along with other Disney Fairies properties launching in the same timeframe, the game represents the growth of a beloved franchise for the company which is sure to be very popular with kids and grown-ups alike."

In Disney Fairies: Tinker Bell for the Nintendo DS, players can use their stylus to fly around the beautiful and always-changing world of Pixie Hollow as Tinker Bell, the iconic Disney Fairy. Players can also fully customize Tinker Bell's wardrobe, choosing from hundreds of different outfits and accessories, and interact with dozens of other Disney Fairies also seen in the all-new movie "Tinker Bell," including Fawn, Iridessa, Rosetta and Silvermist. Along with playing such challenging and fun mini-games as catching dew drops and painting lady bugs, players can make new Fairy friends and trade clothing creations to start style trends. Once a unique clothing item has been created and gifted, witness Pixie Hollow populate with other Fairies wearing the trend.

Because Disney Fairies: Tinker Bell utilizes the DS real-time clock, Pixie Hollow changes with the time of day and the season. Players will witness their birthdays being celebrated, fireflies glowing at night, and their Fairy friends in costume at Halloween. Special items hidden throughout Pixie Hollow contain secret codes which can be used to unlock items in the virtual world on PixieHollow.com, where players can log in, create their own personalized Fairy and help bring about the change of seasons through meeting friends, playing games and collecting items in nature.

Disney Fairies: Tinker Bell also includes access to DGamer, the all new online community exclusively for Disney gamers. Players can create unique 3-D avatars, chat with friends, swap accessories, earn in-game honors and unlock exclusive Disney items during gameplay.

Rated E for Everyone by the Entertainment Software Rating Board (ESRB), Disney Fairies: Tinker Bell is available in stores nationwide today, retailing for $29.99.

For more information, please log on to www.disneyinteractivestudios.com.

Titles, pricing and dates are not final and may be subject to change.

Source: PR Inside

Last Titanic survivor sells mementos

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As a 2-month-old baby, Millvina Dean was wrapped in a sack and lowered into a lifeboat from the deck of the sinking RMS Titanic.

Now, Dean, the last living survivor of the disaster, is selling some of her mementos to help pay her nursing home fees.

Dean's artifacts, including a suitcase given to her family by the people of New York after their rescue, are expected to sell for about STG3,000 ($A7,900) at Saturday's auction in Devizes, western England.

Dean, 96, has lived in a nursing home in the southern English city of Southampton - Titanic's home port - since she broke her hip two years ago.

"I am not able to live in my home any more," Dean was quoted as telling the Southern Daily Echo newspaper. "I am selling it all now because I have to pay these nursing home fees and am selling anything that I think might fetch some money."

Dean's items form part of a sale by Henry Aldridge and Son, an auction house that specialises in Titanic memorabilia.

Auctioneer Andrew Aldridge said the key item was a small wicker suitcase that was filled with clothes and donated to Dean's surviving family members after the disaster.

"They would have carried their little world in this suitcase," Aldridge said.

Dean also is selling letters from the Titanic Relief Fund offering her mother one pound, seven shillings and sixpence a week in compensation.

In 1912, baby Elizabeth Gladys "Millvina" Dean and her family were steerage passengers emigrating to Kansas City, Missouri, aboard the giant cruise liner.

Four days out of port, on the night of April 14, 1912, it hit an iceberg and sank. Billed as "practically unsinkable" by the publicity magazines of the period, the Titanic did not have enough lifeboats for all of 2,200 passengers and crew.

Dean, her mother and 2-year-old brother were among 706 people - mostly women and children - who survived. Her father was among more than 1,500 who died.

Dean did not know she had been aboard the Titanic until she was 8 years old, when her mother, who was about to remarry, told her about her father's death.

She had no memories of the sinking, and said she preferred it that way.

"I wouldn't want to remember, really," she told The Associated Press in 1997.

Dean began to take part in Titanic-related activities in the 1980s, and was active well into her 90s. She visited Belfast to see where the ship was built, attended Titanic conventions around the world - where she was mobbed by autograph seekers - and participated in radio and television documentaries about the sinking.

The last American survivor of the disaster, Lillian Asplund, died in 2006 at the age of 99. Another British survivor, Barbara Joyce West Dainton, died last November at 96.

Aldridge said the "massive interest" in Titanic memorabilia shows no signs of abating. Last year, a collection of items belonging to Asplund sold for more than STG100,000 ($A263,600).

"It's the people, the human angle," Aldridge said. "You had over 2,200 men, women and children on that ship, from John Jacob Astor, the richest person in the world at the time, to a poor Scandinavian family emigrating to the States to start a new life. There were 2,200 stories."


Source: SMH

Titanic: The Artifact Exhibition to Permanently Dock At Luxor Hotel and Casino in Las Vegas December 20th

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Premier Exhibitions, Inc. announced today that Titanic: The Artifact Exhibition will open at Luxor Hotel and Casino in Las Vegas December 20, 2008. Newly designed for Luxor, this blockbuster exhibition brings to life the story of the ill-fated Ship through its authentic artifacts, dramatic room recreations and hands-on interactive experiences.

"With more than 22 million visitors to date, we consistently see how Titanic resonates and touches everyone," states Arnie Geller, Chairman and CEO of Premier Exhibitions, Inc. "Each of us can relate to someone on Titanic and the retelling of her story feeds our curiosity time and again. We are thrilled to be part of the transformation now taking place at Luxor Hotel and Casino and look forward to working with their exceptional staff and management as we present this blockbuster experience."

Felix Rappaport, president and COO of Luxor, said, "The addition of Titanic: The Artifact Exhibition is another new and exciting reason to come visit Luxor. Titanic's maiden voyage is a moment in time that has captivated us for decades, and Premier Exhibitions has done a tremendous job bringing authentic pieces of the story together to recreate this historic event."
As visitors move through expanded Exhibition galleries, they will experience this important historical event, disaster and passenger stories in chronological order: from the construction yards, to her maiden voyage, the fated sinking, to modern day recovery efforts. Each guest will become a passenger and feel the excitement of that April day as they step into authentically re-created first and third class cabins, stroll through the Promenade Deck and stop for a breathtaking glance at the social hub of Titanic -- the Grand Staircase. They will then feel the temperature drop, learn details of the sinking, touch an iceberg and understand the theories of this tragic tale.

Unique to the Exhibition at Luxor are more than 20 never before seen artifacts including gaming chips, passenger personal papers and decorative sections from Titanic's famed Grand Staircase. In addition, the Exhibition will showcase the largest piece of the Titanic ever recovered -- the 'Big Piece.' This 15-ton piece of the Titanic's starboard hull was raised 12,500 feet from the ocean floor and took more than 80 years, and two attempts, to surface.
Providing a striking visual on the Las Vegas Strip, the 30-story Luxor is an architectural wonder boasting one of the world's largest atriums. A multi-phase enhancement plan is revolutionizing the iconic resort and bringing new energy to the south end of the Las Vegas Strip. Luxor is collaborating with industry leaders, like Premier Exhibitions, to bring exciting new entertainment, restaurant and nightclub amenities to the property.

Premier Exhibitions, Inc. is a major provider of museum quality touring exhibitions throughout the world.

Forward-Looking Statements

Certain of the statements contained in this press release contain forward-looking statements that involve a number of risks and uncertainties. Such forward-looking statements are within the meaning of that term in Section 27A of the Securities Act of 1933, as amended, and Section 21E of the Securities Exchange Act of 1934, as amended. Premier Exhibitions, Inc. has based these forward-looking statements on its current expectations and projections about future events, based on the information currently available to it. The forward-looking statements contained in this press release may also include statements relating to Premier Exhibitions' anticipated financial performance, business prospects, new developments, strategies and similar matters. Certain of the factors described in Premier Exhibitions' filings with the Securities and Exchange Commission, including the section of its Annual Report on Form 10-K for the year ended February 28, 2008 titled "Risk Factors," may affect Premier Exhibitions' future results and cause those results to differ materially from those expressed in the forward-looking statements. Premier Exhibitions disclaims any obligation to update any of its forward-looking statements, except as may be required by law.
This news release was distributed by GlobeNewswire, www.globenewswire.com

SOURCE: Premier Exhibitions, Inc.
Premier Exhibitions, Inc.
Investor Relations:
Bud Ingalls, Chief Financial Officer
404-842-2600
bingalls@prxi.com
Media Inquiries:
Katherine Morgenstern, Director of Public Relations
404-842-2600
kmorgenstern@prxi.com

Source: Market Watch

Mark Felt, Watergate's `Deep Throat,' dies at 95

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W. Mark Felt, the former FBI second-in-command who revealed himself as "Deep Throat" 30 years after he tipped off reporters to the Watergate scandal that toppled a president, has died. He was 95.

Felt died Thursday of congestive heart failure in Santa Rosa after several months of failing health, said family friend John D. O'Connor, who wrote the 2005 Vanity Fair article uncovering Felt's secret.

The shadowy central figure in the one of the most gripping political dramas of the 20th century, Felt insisted his alter ego be kept secret when he leaked damaging information about President Richard Nixon and his aides to The Washington Post.

While some — including Nixon and his aides — speculated that Felt was the source who connected the White House to the June 1972 break-in at the headquarters of the Democratic National Committee, he steadfastly denied the accusations until finally coming forward in May 2005.

"I'm the guy they used to call Deep Throat," Felt told O'Connor for the Vanity Fair article, creating a whirlwind of media attention.

The man who had kept his secret for decades, now weakened by a stroke, wasn't doing much talking — he merely waved the media from the front door of his daughter's Santa Rosa home.

Critics, including those who went to prison for the Watergate scandal, called him a traitor for betraying the commander in chief. Supporters hailed him as a hero for blowing the whistle on a corrupt administration trying to cover up attempts to sabotage opponents.

Felt grappled with his place in history, arguing with his children over whether to reveal his identity or to take his secret to the grave, O'Connor said. He agonized about what revealing his identity would do to his reputation. Would he be seen as a turncoat or a man of honor?

Ultimately, his daughter Joan persuaded him to go public; after all, Washington Post reporter Bob Woodward was sure to profit by revealing the secret after Felt died. "We could make at least enough money to pay some bills, like the debt I've run up for the kids' education," she told her father, according to the Vanity Fair article. "Let's do it for the family."

The revelation capped a Washington whodunnit that spanned more than three decades and seven presidents. It was the final mystery of Watergate, the subject of the best-selling book and hit movie "All the President's Men," which inspired a generation of college students to pursue journalism.

It was by chance that Felt came to play a pivotal role in the drama.

Back in 1970, Woodward struck up a conversation with Felt while both men were waiting in a White House hallway. Felt apparently took a liking to the young Woodward, then a Navy courier, and Woodward kept the relationship going, treating Felt as a mentor as he tried to figure out the ways of Washington.

Later, while Woodward and partner Carl Bernstein relied on various unnamed sources in reporting on Watergate, the man their editor dubbed "Deep Throat" helped to keep them on track and confirm vital information. The Post won a Pulitzer Prize for its Watergate coverage.

Within days of the burglary at Watergate that launched the Post's investigative series, Woodward phoned Felt.

"He reminded me how he disliked phone calls at the office but said that the Watergate burglary case was going to `heat up' for reasons he could not explain," Woodward wrote after Felt was named. "He then hung up abruptly."

Felt helped Woodward link former CIA man Howard Hunt to the break-in. He said the reporter could accurately write that Hunt, whose name was found in the address book of one of the burglars, was a suspect. But Felt told him off the record, insisting that their relationship and Felt's identity remain secret.

Worried that phones were being tapped, Felt arranged clandestine meetings worthy of a spy novel. Woodward would move a flower pot with a red flag on his balcony if he needed to meet Felt. The G-man would scrawl a time to meet on page 20 of Woodward's copy of The New York Times and they would rendezvous in a suburban Virginia parking garage in the dead of night.

In the movie, the enduring image of Deep Throat — a name borrowed from a 1972 porn movie — is of a testy, chain-smoking Hal Holbrook telling Woodward, played by Robert Redford, to "follow the money."

In a memoir published in April 2006, Felt said he saw himself as a "Lone Ranger" who could help derail a White House cover-up.

Felt wrote that he was upset by the slow pace of the FBI investigation into the Watergate break-in and believed the press could pressure the administration to cooperate.

"From the start, it was clear that senior administration officials were up to their necks in this mess, and that they would stop at nothing to sabotage our investigation," Felt wrote in "A G-Man's Life: The FBI, `Deep Throat' and the Struggle for Honor in Washington."

Some critics said that Felt, a J. Edgar Hoover loyalist, was bitter at being passed over when Nixon appointed an FBI outsider and confidante, L. Patrick Gray, to lead the FBI after Hoover's death. Gray was later implicated in Watergate abuses.

"We had no idea of his motivations, and even now some of his motivations are unclear," Bernstein said.

Felt wrote that he wasn't motivated by anger. "It is true that I would have welcomed an appointment as FBI director when Hoover died. It is not true that I was jealous of Gray," he wrote.

For his part, Holbrook responded to the news of Felt's identity by commenting: "He was doing it because there was a higher purpose involved. ... The important thing was not who it was, but why he did it. It's called morality. That is something that is not very popular today."

Felt was born in Twin Falls, Idaho, and worked for an Idaho senator during graduate school. After law school at George Washington University he spent a year at the Federal Trade Commission. Felt joined the FBI in 1942 and worked as a Nazi hunter during World War II.

Ironically, while providing crucial information to the Post, Felt also was assigned to ferret out the newspaper's source. The investigation never went anywhere, but plenty of people, including those in the White House at the time, guessed that Felt, who was leading the investigation into Watergate, may have been acting as a double agent.

The Watergate tapes captured White House chief of staff H.R. Haldeman telling Nixon that Felt was the source, but they were afraid to stop him.

Nixon asks: "Somebody in the FBI?"

Haldeman: "Yes, sir. Mark Felt ... If we move on him, he'll go out and unload everything. He knows everything that's to be known in the FBI."

Felt left the FBI in 1973 for the lecture circuit. Five years later he was indicted on charges of authorizing FBI break-ins at homes associated with suspected bombers from the 1960s radical group the Weather Underground. President Ronald Reagan pardoned Felt in 1981 while the case was on appeal — a move applauded by Nixon.

Woodward and Bernstein said they wouldn't reveal the source's identity until he or she died, and finally confirmed Felt's role only after he came forward.

"People will debate for a long time whether I did the right thing by helping Woodward," Felt wrote in his memoir. "The bottom line is that we did get the whole truth out, and isn't that what the FBI is supposed to do?"

Source: The Associated Press

Types of Triangles

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ABC.

In Euclidean geometry any three non-collinear points determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space).

Types of triangles

By relative lengths of sides

Triangles can be classified according to the relative lengths of their sides:

  • In an equilateral triangle, all sides are of equal length. An equilateral triangle is also an equiangular polygon, i.e. all its internal angles are equal—namely, 60°; it is a regular polygon.[1]
  • In an isosceles triangle, two sides are of equal length (originally and conventionally limited to exactly two).[2] An isosceles triangle also has two equal angles: the angles opposite the two equal sides.
  • In a scalene triangle, all sides have different lengths. The internal angles in a scalene triangle are all different.[3]
Equilateral Triangle Isosceles triangle Scalene triangle

By internal angles

Triangles can also be classified according to their internal angles, described below using degrees of arc:

  • A right triangle (or right-angled triangle, formerly called a rectangled triangle) has one 90° internal angle (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side in the right triangle. The other two sides are the legs or catheti (singular: cathetus) of the triangle. Right triangles conform to the Pythagorean theorem, wherein the sum of the squares of the two legs is equal to the square of the hypotenuse, i.e., a2 + b2 = c2, where a and b are the legs and c is the hypotenuse. See also Special right triangles
  • An oblique triangle has no internal angle equal to 90°.
  • An obtuse triangle is an oblique triangle with one internal angle larger than 90° (an obtuse angle).
  • An acute triangle is an oblique triangle with internal angles all smaller than 90° (three acute angles). An equilateral triangle is an acute triangle, but not all acute triangles are equilateral triangles.
Right triangle Obtuse triangle Acute triangle
Right Obtuse Acute

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Oblique

Basic facts

Elementary facts about triangles were presented by Euclid in books 1-4 of his Elements around 300 BCE. A triangle is a polygon and a 2-simplex (see polytope). All triangles are two-dimensional.

The angles of a triangle in Euclidean space always add up to 180 degrees. An exterior angle of a triangle (an angle that is adjacent and supplementary to an internal angle) is always equal to the two angles of a triangle that it is not adjacent/supplementary to; this is the exterior angle theorem. Like all convex polygons, the exterior angles of a triangle add up to 360 degrees.

The sum of the lengths of any two sides of a triangle always exceeds the length of the third side. That is the triangle inequality. (In the special case of equality, two of the angles have collapsed to size zero, and the triangle has degenerated to a line segment.)

Two triangles are said to be similar if and only if the angles of one are equal to the corresponding angles of the other. In this case, the lengths of their corresponding sides are proportional. This occurs for example when two triangles share an angle and the sides opposite to that angle are parallel.

A few basic postulates and theorems about similar triangles:

  • Two triangles are similar if at least two corresponding angles are equal.
  • If two corresponding sides of two triangles are in proportion, and their included angles are equal, the triangles are similar.
  • If three sides of two triangles are in proportion, the triangles are similar.

For two triangles to be congruent, each of their corresponding angles and sides must be equal (6 total). A few basic postulates and theorems about congruent triangles:

  • SAS Postulate: If two sides and the included angles of two triangles are correspondingly equal, the two triangles are congruent.
  • SSS Theorem: If every side of two triangles are correspondingly equal, the triangles are congruent.
  • ASA Theorem: If two angles and the included sides of two triangles are correspondingly equal, the two triangles are congruent.
  • AAS Theorem: If two angles and any side of two triangles are correspondingly equal, the two triangles are congruent.
  • Hypotenuse-Leg Theorem: If the hypotenuses and one leg of two right triangles are correspondingly equal, the triangles are congruent.
  • Hypotenuse-Angle Theorem: If the hypotenuse and an acute angle of one right triangle are congruent to a hypotenuse and an acute angle of another right triangle, then the triangles are congruent
  • Side-Side-Angle (or Angle-Side-Side) condition: if two sides and an angle that isn't included of two triangles are equal, then if the angle is obtuse, the opposite side is longer than the adjacent, or the opposite side is equal to the sine of the angle times the adjacent side, the triangles are congruent.

Using right triangles and the concept of similarity, the trigonometric functions sine and cosine can be defined. These are functions of an angle which are investigated in trigonometry.

In Euclidean geometry, the sum of the internal angles of a triangle is equal to 180°. This allows determination of the third angle of any triangle as soon as two angles are known.

The Pythagorean theorem

A central theorem is the Pythagorean theorem, which states in any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two other sides. If the hypotenuse has length c, and the legs have lengths a and b, then the theorem states that

a^2 + b^2=c^2.   \,

The converse is true: if the lengths of the sides of a triangle satisfy the above equation, then the triangle is a right triangle.

Some other facts about right triangles:

  • The acute angles of a right triangle are complementary.
  • If the legs of a right triangle are equal, then the angles opposite the legs are equal, acute and complementary, and thus are both 45 degrees. By the Pythagorean theorem, the length of the hypotenuse is the length of a leg times the square root of two.
  • In a 30-60 right triangle, in which the acute angles measure 30 and 60 degrees, the hypotenuse is twice the length of the shorter side.
  • In all right triangles, the median on the hypotenuse is half of the hypotenuse.

For all triangles, angles and sides are related by the law of cosines and law of sines.

Points, lines and circles associated with a triangle

There are hundreds of different constructions that find a special point inside a triangle, satisfying some unique property: see the references section for a catalogue of them. Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point: an important tool for proving the existence of these is Ceva's theorem, which gives a criterion for determining when three such lines are concurrent. Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points are collinear: here Menelaus' theorem gives a useful general criterion. In this section just a few of the most commonly-encountered constructions are explained.

The circumcenter is the center of a circle passing through the three vertices of the triangle.

A perpendicular bisector of a triangle is a straight line passing through the midpoint of a side and being perpendicular to it, i.e. forming a right angle with it. The three perpendicular bisectors meet in a single point, the triangle's circumcenter; this point is the center of the circumcircle, the circle passing through all three vertices. The diameter of this circle can be found from the law of sines stated above.

Thales' theorem implies that if the circumcenter is located on one side of the triangle, then the opposite angle is a right one. More is true: if the circumcenter is located inside the triangle, then the triangle is acute; if the circumcenter is located outside the triangle, then the triangle is obtuse.

The intersection of the altitudes is the orthocenter.

An altitude of a triangle is a straight line through a vertex and perpendicular to (i.e. forming a right angle with) the opposite side. This opposite side is called the base of the altitude, and the point where the altitude intersects the base (or its extension) is called the foot of the altitude. The length of the altitude is the distance between the base and the vertex. The three altitudes intersect in a single point, called the orthocenter of the triangle. The orthocenter lies inside the triangle if and only if the triangle is acute. The three vertices together with the orthocenter are said to form an orthocentric system.

The intersection of the angle bisectors finds the center of the incircle.

An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half. The three angle bisectors intersect in a single point, the incenter, the center of the triangle's incircle. The incircle is the circle which lies inside the triangle and touches all three sides. There are three other important circles, the excircles; they lie outside the triangle and touch one side as well as the extensions of the other two. The centers of the in- and excircles form an orthocentric system.

The intersection of the medians is the centroid.

A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas. The three medians intersect in a single point, the triangle's centroid. The centroid of a stiff triangular object (cut out of a thin sheet of uniform density) is also its center of gravity: the object can be balanced it on its centroid. The centroid cuts every median in the ratio 2:1, i.e. the distance between a vertex and the centroid is twice the distance between the centroid and the midpoint of the opposite side.

Nine-point circle demonstrates a symmetry where six points lie on the edge of the triangle.

The midpoints of the three sides and the feet of the three altitudes all lie on a single circle, the triangle's nine-point circle. The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the orthocenter. The radius of the nine-point circle is half that of the circumcircle. It touches the incircle (at the Feuerbach point) and the three excircles.

Euler's line is a straight line through the centroid (orange), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red).

The centroid (yellow), orthocenter (blue), circumcenter (green) and barycenter of the nine-point circle (red point) all lie on a single line, known as Euler's line (red line). The center of the nine-point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter.

The center of the incircle is not in general located on Euler's line.

If one reflects a median at the angle bisector that passes through the same vertex, one obtains a symmedian. The three symmedians intersect in a single point, the symmedian point of the triangle.

Computing the area of a triangle

Calculating the area of a triangle is an elementary problem encountered often in many different situations. The best known and simplest formula is:

S=\frac{1}{2}bh

where S is area, b is the length of the base of the triangle, and h is the height or altitude of the triangle. The term 'base' denotes any side, and 'height' denotes the length of a perpendicular from the point opposite the side onto the side itself.

Although simple, this formula is only useful if the height can be readily found. For example, the surveyor of a triangular field measures the length of each side, and can find the area from his results without having to construct a 'height'. Various methods may be used in practice, depending on what is known about the triangle. The following is a selection of frequently used formulae for the area of a triangle.[4]

Using vectors

The area of a parallelogram can be calculated using vectors. Let vectors AB and AC point respectively from A to B and from A to C. The area of parallelogram ABDC is then |{AB}\times{AC}|, which is the magnitude of the cross product of vectors AB and AC. |{AB}\times{AC}| is equal to |{h}\times{AC}|, where h represents the altitude h as a vector.

The area of triangle ABC is half of this, or S = \frac{1}{2}|{AB}\times{AC}|.

The area of triangle ABC can also be expressed in terms of dot products as follows:

\frac{1}{2} \sqrt{(\mathbf{AB} \cdot \mathbf{AB})(\mathbf{AC} \cdot \mathbf{AC}) -(\mathbf{AB} \cdot \mathbf{AC})^2} =\frac{1}{2} \sqrt{ |\mathbf{AB}|^2 |\mathbf{AC}|^2 -(\mathbf{AB} \cdot \mathbf{AC})^2} \, .

Using trigonometry

The height of a triangle can be found through an application of trigonometry. Using the labelling as in the image on the left, the altitude is h = a sin γ. Substituting this in the formula S = ½bh derived above, the area of the triangle can be expressed as:

S =  \frac{1}{2}ab\sin \gamma = \frac{1}{2}bc\sin \alpha  = \frac{1}{2}ca\sin \beta.

Furthermore, since sin α = sin (π - α) = sin (β + γ), and similarly for the other two angles:

S = \frac{1}{2}ab\sin (\alpha+\beta) = \frac{1}{2}bc\sin (\beta+\gamma) = \frac{1}{2}ca\sin (\gamma+\alpha).

Using coordinates

If vertex A is located at the origin (0, 0) of a Cartesian coordinate system and the coordinates of the other two vertices are given by B = (xB, yB) and C = (xC, yC), then the area S can be computed as ½ times the absolute value of the determinant

S=\frac{1}{2}\left|\det\begin{pmatrix}x_B & x_C \\ y_B & y_C \end{pmatrix}\right| = \frac{1}{2}|x_B y_C - x_C y_B|.

For three general vertices, the equation is:

S=\frac{1}{2} \left| \det\begin{pmatrix}x_A & x_B & x_C \\  y_A & y_B & y_C \\ 1 & 1 & 1\end{pmatrix} \right| = \frac{1}{2} \big| x_A y_C - x_A y_B + x_B y_A - x_B y_C + x_C y_B - x_C y_A \big|
S= \frac{1}{2} \big| (x_C - x_A) (y_B - y_A) - (x_B - x_A) (y_C - y_A) \big|.

In three dimensions, the area of a general triangle {A = (xA, yA, zA), B = (xB, yB, zB) and C = (xC, yC, zC)} is the Pythagorean sum of the areas of the respective projections on the three principal planes (i.e. x = 0, y = 0 and z = 0):

S=\frac{1}{2} \sqrt{ \left( \det\begin{pmatrix} x_A & x_B & x_C \\ y_A & y_B & y_C \\ 1 & 1 & 1 \end{pmatrix} \right)^2 + \left( \det\begin{pmatrix} y_A & y_B & y_C \\ z_A & z_B & z_C \\ 1 & 1 & 1 \end{pmatrix} \right)^2 + \left( \det\begin{pmatrix} z_A & z_B & z_C \\ x_A & x_B & x_C \\ 1 & 1 & 1 \end{pmatrix} \right)^2 }.

Using Heron's formula

The shape of the triangle is determined by the lengths of the sides alone. Therefore the area S also can be derived from the lengths of the sides. By Heron's formula:

S = \sqrt{s(s-a)(s-b)(s-c)}

where s = ½ (a + b + c) is the semiperimeter, or half of the triangle's perimeter.

Three equivalent ways of writing Heron's formula are

 S = \frac{1}{4} \sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}
 S = \frac{1}{4} \sqrt{2(a^2b^2+a^2c^2+b^2c^2)-(a^4+b^4+c^4)}
 S = \frac{1}{4} \sqrt{(a+b-c) (a-b+c) (-a+b+c) (a+b+c)}.

Computing the sides and angles

In general, there are various accepted methods of calculating the length of a side or the size of an angle. Whilst certain methods may be suited to calculating values of a right-angled triangle, others may be required in more complex situations.

The sine and cosine rules

The law of sines, or sine rule[5], states that the ratio of the length of side a to the sine of its corresponding angle α is equal to the ratio of the length of side b to the sine of its corresponding angle β.

\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}

The law of cosines, or cosine rule, connects the length of an unknown side of a triangle to the length of the other sides and the angle opposite to the unknown side. As per the law:

For a triangle with length of sides a, b, c and angles of α, β, γ respectively, given two known lengths of a triangle a and b, and the angle between the two known sides γ (or the angle opposite to the unknown side c), to calculate the third side c, the following formula can be used:

c^2\ = a^2 + b^2 - 2ab\cos(\gamma) \implies b^2\ = a^2 + c^2 - 2ac\cos(\beta) \implies a^2\ = b^2 + c^2 - 2bc\cos(\alpha)

Trigonometric ratios in right triangles

In right triangles, the trigonometric ratios of sine, cosine and tangent can be used to find unknown angles and the lengths of unknown sides. The sides of the triangle are known as follows:

  • The hypotenuse is the side opposite the right angle, or defined as the longest side of a right-angled triangle, in this case h.
  • The opposite side is the side opposite to the angle we are interested in, in this case a.
  • The adjacent side is the side that is in contact with the angle we are interested in and the right angle, hence its name. In this case the adjacent side is b.

Sine, cosine and tangent

The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case

\sin A = \frac {\textrm{opposite}} {\textrm{hypotenuse}} = \frac {a} {h}\,.

Note that this ratio does not depend on the particular right triangle chosen, as long as it contains the angle A, since all those triangles are similar.

The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In our case

\cos A = \frac {\textrm{adjacent}} {\textrm{hypotenuse}} = \frac {b} {h}\,.

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In our case

\tan A = \frac {\textrm{opposite}} {\textrm{adjacent}} = \frac {a} {b}\,.

The acronym "SOHCAHTOA" is a useful mnemonic for these ratios.

Inverse functions

The inverse trigonometric functions can be used to calculate the internal angles for a right angled triangle with the length of any two sides.

Arcsin can be used to calculate an angle from the length of the opposite side and the length of the hypotenuse.

\theta = \arcsin \left( \frac{\text{opposite}}{\text{hypotenuse}} \right)

Arccos can be used to calculate an angle from the length of the adjacent side and the length of the hypontenuse.

\theta = \arccos \left( \frac{\text{adjacent}}{\text{hypotenuse}} \right)

Arctan can be used to calculate an angle from the length of the opposite side and the length of the adjacent side.

\theta = \arctan \left( \frac{\text{opposite}}{\text{adjacent}} \right)

Non-planar triangles

A non-planar triangle is a triangle which is not contained in a (flat) plane. Examples of non-planar triangles in noneuclidean geometries are spherical triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry.

While all regular, planar (two dimensional) triangles contain angles that add up to 180°, there are cases in which the angles of a triangle can be greater than or less than 180°. In curved figures, a triangle on a negatively curved figure ("saddle") will have its angles add up to less than 180° while a triangle on a positively curved figure ("sphere") will have its angles add up to more than 180°. Thus, if one were to draw a giant triangle on the surface of the Earth, one would find that the sum of its angles were greater than 180°.


Source: Wikipedia