Triangle theorems and special segments

Several key theorems describe triangles. The Isosceles Triangle Theorem says the base angles opposite a triangle's two congruent sides are congruent, and its converse is also true. An exterior angle of a triangle equals the sum of the two remote interior angles. A midsegment, joining the midpoints of two sides, is parallel to the third side and half as long, so a third side of 14 gives a midsegment of 7. A median joins a vertex to the midpoint of the opposite side; the three medians meet at the centroid, which lies two thirds of the way from each vertex to the opposite midpoint. By the Perpendicular Bisector Theorem, every point on a segment's perpendicular bisector is equidistant from its endpoints, so the three perpendicular bisectors of a triangle meet at a point, the circumcenter, equally distant from all three vertices; the angle bisectors meet at the incenter, equally distant from all three sides. Finally, the Triangle Inequality Theorem says any two sides must add up to more than the third, so lengths 3, 4, and 8 cannot form a triangle.

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