parmenides51 wrote: Let $AH_a $ and $BH_b$ be the altitudes of triangle $ABC$. Points $P$ and $Q$ are the projections of $H_a$ to $AB$ and $AC$. Prove that line $PQ $ bisects segment $H_aH_b$. (A.Akopjan, K.Savenkov) By direct counting of the angles: $$\angle QH_bH_a=\angle AH_aH_b=\angle H_bBP=\angle H_bQP,$$hence if $X=H_aH_b\cap PQ$, then $H_bX=XQ$. As $\angle H_bQH_a=90^\circ$, then $H_bX=XQ=XH_a$.