Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 6.2 National Olympiad Prove it JBMO

Problem:

The point $P$ is outside of the circle $\Omega$ . Two tangent lines, passing from the point $P$ , touch the circle $\Omega$ at the points $A$ and $B$ . The median $A M$ , $M \in (B P)$ , intersects the circle $\Omega$ at the point $C$ and the line $P C$ intersects again the circle $\Omega$ at the point $D$ . Prove that the lines $A D$ and $B P$ are parallel.

Solution

Solution:

Since $\angle B A C = \angle B A M = \angle M B C$ , we have $\triangle M A B \cong \triangle M B C$ .

We obtain $\frac{M A}{M B} = \frac{M B}{M C} = \frac{A B}{B C}$ . The equality $\quad M B = M P$ implies $\frac{M A}{M P} = \frac{M P}{M C}$ and $\angle P M C \equiv \angle P M A$ gives the relation $\triangle P M A \cong \triangle C M P$ . It follows that $\angle B P D \equiv \angle M P C \equiv \angle M A P \equiv \angle C A P \equiv \angle C D A \equiv \angle P D A$ . So, the lines $A D$ and $B P$ are parallel.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.