 | Triangle: Encyclopedia II - Triangle - Basic facts
Triangle - 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).
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.
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 the remainder we will consider a triangle with vertices A, B and C, angles α, β and γ and sides a, b and c. The side a is opposite to the vertex A and angle α and analogously for the other sides.
In Euclidean geometry, the sum of the angles α + β + γ is equal to two right angles (180° or π radians). This allows determination of the third angle of any triangle as soon as two angles are known.
A central theorem is the Pythagorean theorem stating that in any right triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides. If side C is the hypotenuse, we can write this as
This means that knowing the lengths of two sides of a right triangle is enough to calculate the length of the third—something unique to right triangles. The Pythagorean theorem can be generalized to the law of cosines:
which is valid for all triangles, even if γ is not a right angle. The law of cosines can be used to compute the side lengths and angles of a triangle as soon as all three sides or two sides and an enclosed angle are known.
The law of sines states
where d is the diameter of the circumcircle (the circle which passes through all three points of the triangle). The law of sines can be used to compute the side lengths for a triangle as soon as two angles and one side are known. If two sides and an unenclosed angle is known, the law of sines may also be used; however, in this case there may be zero, one or two solutions.
There are two special right triangles that appear commonly in geometry. The so-called "45-45-90 triangle" has angles with those angle measures and the ratio of its sides is :. The "30-60-90 triangle" has sides in the ratio of .
Other related archives300 BCE, Cartesian coordinate system, Ceva's theorem, Elements, Euclid, Euler's line, Feuerbach point, Heron's formula, Menelaus' theorem, Pythagorean theorem, Thales' theorem, absolute value, acute angles, altitude, angle, angle bisector, angles, area, center of gravity, centroid, circle, circumcenter, circumcircle, collinear, concurrent, cosine, cross product, degrees, determinant, excircles, geometry, hyperbolic geometry, hyperbolic triangles, incircle, law of cosines, law of sines, line segments, median, nine-point circle, obtuse angle, orthocenter, orthocentric system, parallelogram, perpendicular bisector, plane, polygon, polytope, proportional, regular polygon, right angle, similar, simplex, sine, special right triangles, spherical geometry, spherical triangles, straight, symmedian, symmedian point, trigonometric functions, trigonometry, vectors, vertices
 Adapted from the Wikipedia article "Basic facts", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
