Geometry: Theorems: Theorems for Segments within Triangles

Geometry: Theorems Theorems for Segments within Triangles

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Midsegments of a Triangle

The midsegment of a triangle is a segment whose endpoints are both midpoints of sides. Every triangle has three midsegments. The midsegment of a triangle is always parallel to the third side (the side whose midpoint it doesn't include), and half as long as the third side.

Angle Bisectors of Triangles

The angle bisectors of a triangle intersect each other at a point called the incircle of the triangle. The incircle of a triangle is the same as the center of a circle inscribed in a triangle. Every triangle can have exactly one inscribed circle, whose center is the incircle of the triangle, which is the point at which the angle bisectors of the triangle intersect. The incircle, then, is equidistant from the three sides of the triangle--a property that results from the inherent congruency of the radii of a circle.

Figure %: A triangle's angle bisectors and incircle

Another property of angle bisectors has to do with the side opposite the bisected angle. An angle bisector divides the side opposite the bisected angle into two segments that are of the same proportion as the other two sides. For example, in triangle ABC above, let the angle at vertex A be bisected, and let the bisector intersect BC at point D. BD/DC = BA/CA.

Perpendicular Bisectors of Triangles

The three perpendicular bisectors of a triangle intersect at one point called the circumcenter of a triangle. The circumcenter is the center of the circle circumscribed about the triangle and is equidistant from all the vertices of the triangle. In this case the perpendicular bisectors of the sides of the triangles are lines, not segments. Therefore, the circumcenter of a triangle does not necessarily exist in the interior of the triangle. Often the perpendicular bisectors of a triangle intersect outside the triangle.