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Chapter 3: Polygons and Their Relations

# Triangles

Triangles are polygons with three sides and three angles. All three angles of a triangle add up to 180 degree, and any two sides' sum is larger than the third side. There are three types of triangles: acute triangles, right triangles, and obtuse triangles. The triangles can also be classified into four types: equiangular triangle, equilateral triangle, isosceles triangles, and scalene triangles. All triangles have medians, which are segments whose endpoints are a vertex of the triangle and the midpoint of the opposite side, and altitudes, which are segments from a vertex of the triangle perpendicular to the line containing the opposite side. The altitude from the vertex angle to the base of an isosceles triangle is also the triangle's median. Here are some additional concepts about triangles:

If two sides of a triangle are congruent, then the angles opposite these sides are congruent.

If a triangle is equilateral, then it is also equiangular, and the measure of each angle is 60

If two angles of a triangle are congruent, then the sides opposite these angles are congruent.

If a triangle is equiangular, then it is also equilateral.

The segment joining the midpoints of two sides of a triangle is parallel to the third side, and its length is half the length of the third side.

The circumcenter of a triangle is the intersection of the perpendicular bisectors of the sides of the triangle.

The incenter of a triangle is the intersection of the bisectors of the angles of the triangle.

The orthocenter of a triangle is the intersection of the lines containing the altitudes of the triangle.

The centroid of a triangle is the point of intersection of the medians.

The perpendicular bisectors of the sides of a triangle are congruent at a point equidistant from the vertices of the triangle.

The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle.

The lines containing the altitudes of a triangle are concurrent.

Two medians of a triangle intersect at a point two-thirds of the distance from each vertex to the midpoint of the opposite side.

The medians of a triangle are concurrent at a point that is two-thirds the distance from each vertex to the midpoint of the opposite side.

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