Worked problem — the bisector of the isosceles triangle

Statement

In an isosceles triangle the bisector of the apex angle is also an altitude and a median. Prove it, following the four moves.

Solution

Understand. Triangle $ABC$ with $AB\cong AC$; bisector $AM$ of the angle $\widehat{A}$, $M\in BC$. Twofold thesis: $AM\perp BC$ and $BM\cong MC$ (see understanding a geometry problem).

Search for a route. Reasoning backwards, it is enough to prove that the two triangles $ABM$ and $ACM$ are congruent: from there both theses follow (see searching for the route).

Carry out. In triangles $ABM$ and $ACM$: $AB\cong AC$ (hypothesis), $\widehat{BAM}\cong\widehat{CAM}$ (bisector), $AM$ common. By the first criterion (side–angle–side) $\triangle ABM\cong\triangle ACM$. It follows that $BM\cong CM$ (so $AM$ is a median) and $\widehat{AMB}\cong\widehat{AMC}$; being supplementary, they each equal $\tfrac{\pi}{2}$ (so $AM$ is an altitude). $\blacksquare$

Review. The “isosceles” hypothesis is essential: in a scalene triangle the bisector does not, in general, fall on the midpoint of the opposite side (see reviewing the proof).

Links

Topics: Working method
Concepts: Backward reasoning · Problem solving
Skills: Proving · Synthetic geometry
Exercise type: Proof