Question 61
A triangle $ABC$ is given, right-angled at $A$ and with $AC$ the longer leg; let $M$ be the midpoint of $BC$, $N$ the reflection of $A$ in $BC$, and $O$ the intersection of the perpendicular to $MN$ through $N$ with the line containing $BC$.
a) Prove that the angle $OMN$ is twice the angle $ACB$.
b) Prove that the ratio between the areas of $MNO$ and $ABC$ equals one quarter of the ratio between the lengths of $BC$ and $HM$, where $H$ is the foot of the altitude to the hypotenuse of $ABC$.
a) Prove that the angle $OMN$ is twice the angle $ACB$.
b) Prove that the ratio between the areas of $MNO$ and $ABC$ equals one quarter of the ratio between the lengths of $BC$ and $HM$, where $H$ is the foot of the altitude to the hypotenuse of $ABC$.