Example — reviewing the angle-bisector proof

Example — Reviewing the angle-bisector proof

Check: let us go back through the proof. We used: $AB\cong AC$ (isosceles), $\widehat{BAM}\cong\widehat{CAM}$ (bisector), $AM$ common. Three congruent elements, one of which is an angle included between two sides: the first criterion. ✓

Reflection: the key idea was to “split” the large triangle into two triangles and prove their congruence. It is a technique that will come back many times.

Extension: and what if the triangle were not isosceles? Would the bisector still be a median? And an altitude? Trying with a scalene triangle one sees at once that the answer is no: the bisector meets the opposite side at a point which, in general, is not its midpoint (cf. the angle-bisector theorem). So the “isosceles” hypothesis is essential, not decorative.

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Topics: Working method
Concepts: Checking the result
Skills: Proving · Synthetic geometry