Question 41
The acute-angled triangle $ABC$ has circumcentre $O$ and orthocentre $H$. The perpendicular bisector of $BH$ meets $AB$ at $Q$, the perpendicular bisector of $CH$ meets $AC$ at $P$ and these two bisectors meet at $R$.
a. Prove that the circumcircles of $AOP$, $POR$, $ROQ$ and $QOA$ have equal radii. When are they equal in radius to the circumcircle of triangle $ABC$?
b. Prove that if $PQ$ passes through $O$, then it also passes through $H$. What can be said about $\angle BAC$ in this case?
a. Prove that the circumcircles of $AOP$, $POR$, $ROQ$ and $QOA$ have equal radii. When are they equal in radius to the circumcircle of triangle $ABC$?
b. Prove that if $PQ$ passes through $O$, then it also passes through $H$. What can be said about $\angle BAC$ in this case?