Example — finding the path, the isosceles triangle

Example — Finding the path: the isosceles triangle

To prove that the bisector $AM$ is also an altitude and a median, which path do we choose?

Consider the two small triangles $ABM$ and $ACM$ into which $AM$ divides the triangle. If we prove that they are congruent, then automatically:

• $BM\cong MC$ (corresponding sides) $\implies$ $AM$ is a median;
• $\widehat{AMB}\cong\widehat{AMC}$ and, being supplementary, they are both right angles $\implies$ $AM$ is an altitude.

So the path is: prove the congruence of the two triangles. A single objective instead of two. This is a typical “backward reasoning” move.

Links

Topics: Working method
Concepts: Backward reasoning · Solving strategy
Skills: Proving · Synthetic geometry