Facts about Triangle

Thales' theorem implies that if the circumcenter is located on one side of the triangle, then the opposite angle is a right one.

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

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.

Elementary facts about triangles were presented by Euclid in books 1-4 of his Elements around 300 B.C.E.

An altitude of a triangle is a straight line through a vertex and perpendicular to (i.e.

A triangle is a polygon and a 2-simplex (see polytope).

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.

The orthocenter lies inside the triangle if and only if the triangle is acute.

A triangle is one of the basic shapes of geometry: a polygon with three corners (or vertices) and three sides (or edges) that are straight line segments.

Two triangles are said to be similar if and only if the angles of one are equal to the corresponding angles of the other.

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

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.

Other approaches exist, depending on what is known about the triangle.

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.

What follows is a selection of frequently used formulae for the area of a triangle.

The altitude of a triangle can be found through an application of trigonometry.

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

The three altitudes intersect in a single point, called the orthocenter of the triangle.

Using right triangles and the concept of similarity, the trigonometric functions sine and cosine can be defined.

Calculating the area of a triangle is an elementary problem encountered often in many different situations.

The incircle is the circle which lies inside the triangle and touches all three sides.

The three medians intersect in a single point, the triangle's centroid.

where is area, is the length of the base of the triangle, and is the height or altitude of the triangle.

A non-planar triangle is a triangle which is not contained in a (flat) plane.

The sum of the lengths of any two sides of a triangle always exceeds the length of the third side.

Like all convex polygons, the exterior angles of a triangle add up to 360 degrees.

An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half.

The angles of a triangle add up to 180 degrees.

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.

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 angle bisectors intersect in a single point, the incenter, the center of the triangle's incircle.

A perpendicular bisector of a triangle is a straight line passing through the midpoint of a side and being perpendicular to it (i.e.

The three symmedians intersect in a single point, the symmedian point of the triangle.

The shape of the triangle is determined by the lengths of the sides alone.

Calculating the area of a triangle is an elementary problem encountered often in many different situations.

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.

A triangle is one of the basic shapes of geometry: a polygon with three corners (or vertices) and three sides (or edges) that are straight line segments.