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Two circles and a chord

Source: 44th International Tournament of Towns, Senior A-Level P3, Fall 2022

February 16, 2023
geometryTournament of Towns

Problem Statement

Consider two concentric circles Ω\Omega and ω\omega. Chord ADAD of the circle Ω\Omega is tangent to ω\omega. Inside the minor disk segment ADAD of Ω\Omega, an arbitrary point PP{} is selected. The tangent lines drawn from the point PP{} to the circle ω\omega intersect the major arc ADAD of the circle Ω\Omega at points BB{} and CC{}. The line segments BDBD and ACAC intersect at the point QQ{}. Prove that the line segment PQPQ passes through the midpoint of line segment ADAD.
Note. A circle together with its interior is called a disk, and a chord XYXY of the circle divides the disk into disk segments, a minor disk segment XYXY (the one of smaller area) and a major disk segment XYXY.
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