Question 3
In the acute-angled triangle $ABC$, $AH$ is the longest altitude ($H$ lies on $BC$), $M$ is the midpoint of $AC$, and $CD$ is an angle bisector (with $D$ on $AB$).
a. If $AH \leq BM$, prove that the angle $ABC \leq 60^\circ$.
b. If $AH = BM = CD$, prove that $ABC$ is equilateral.
a. If $AH \leq BM$, prove that the angle $ABC \leq 60^\circ$.
b. If $AH = BM = CD$, prove that $ABC$ is equilateral.