Isosceles & Equilateral Triangles

In an isosceles triangle the two equal sides are the legs, the third is the base, and the two angles at the base are congruent.

The Base Angles Theorem: if two sides are congruent, the angles opposite them are congruent.

Its converse is also true: if two angles are congruent, the sides opposite them are congruent. This is how you prove a triangle is isosceles.

Equilateral triangles are the special case. All three sides equal means all three angles equal, and since they sum to 180°, each is 60°.

One more useful fact: in an isosceles triangle, the segment from the apex to the midpoint of the base is simultaneously the median, the altitude, the angle bisector and the perpendicular bisector. All four coincide.

The Isosceles Triangle Theorem states that if AB ≅ AC then ∠B ≅ ∠C. The classical proof draws the bisector of the apex angle and applies SAS, then CPCTC.

Its converse — congruent base angles imply congruent opposite sides — is proved similarly and is what licenses concluding "therefore the triangle is isosceles" from angle information alone.

Corollaries: a triangle is equilateral if and only if it is equiangular, and each angle of an equilateral triangle measures 60°.

The coincidence of the four special segments from the apex is a genuinely useful fact, because it means a single auxiliary line can be justified as whichever of the four the proof needs. The converse direction is also true and gives a test: if a triangle's median from a vertex is also an altitude, the triangle is isosceles.

This coincidence is the reason equilateral triangles have their centroid, circumcentre, incentre and orthocentre all at the same point, which is not the case for any other triangle.