Problem:
H is the orthocenter of acute triangle ABC, from A, draw the two tangent lines AP and AQ of the circle whose diameter is BC, the points of tangency are P, Q respectively. Prove: P, H, Q are collinear.
- Let $\displaystyle \omega$ be the circle with diameter $\displaystyle BC$
- Let $\displaystyle B'$ & $\displaystyle C'$ be the projections of $\displaystyle B$ & $\displaystyle C$ to $\displaystyle \overline{AC}$ & $\displaystyle \overline{AB}$ respectively.
- In the cyclic quadrilateral $\displaystyle BCB'C'$ with $\displaystyle A\in\overline{BC'}\cap\overline{CB'}$ & $\displaystyle H\in\overline{BB'}\cap\overline{CC'}$ yields that $\displaystyle H$ is on the polar line of $\displaystyle A\,\blacksquare$