Proving 2 lines are parallel.

Question

The point $P$ is situated outside of the circle $Ω$. Two lines, passing through $P$, are tangent to $Ω$ in $A$ and $B$. The median $AM$ of the $\triangle PAB$, with $M$ situated on $BP$, intersects the circle $Ω$ at the point $C$. The line $PC$ intersects again the circle $Ω$ at the point $D$. Prove that the lines $AD$ and $BP$ are parallel.

Answer 1

• Since $MP^2 = MB^2 = MC \cdot MA$ we conclude $BP$ is tangent to circle $(ACP)$.
• From tangent-chord property we have (use it twice): $$\angle MPC = \angle CAP = \angle ADC$$ and so $AD||BP$.