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Equal angles in a donut

Source: 2022 China TST, Test 3 P1

April 30, 2022
geometrycirclesequal angles

Problem Statement

Given two circles ω1\omega_1 and ω2\omega_2 where ω2\omega_2 is inside ω1\omega_1. Show that there exists a point PP such that for any line ℓ\ell not passing through PP, if ℓ\ell intersects circle ω1\omega_1 at A,BA,B and ℓ\ell intersects circle ω2\omega_2 at C,DC,D, where A,C,D,BA,C,D,B lie on ℓ\ell in this order, then ∠APC=∠BPD\angle APC=\angle BPD.
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