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fixed angle wanted, 2 mixtlinears created by a perpendicular on semicircle

Source: 2020 Austrian Federal Competition For Advanced Students, Part 2, p5

November 22, 2020
geometrymixtilinearfixedfixed anglesemicircle

Problem Statement

Let hh be a semicircle with diameter ABAB. Let PP be an arbitrary point inside the diameter ABAB. The perpendicular through PP on ABAB intersects hh at point CC. The line PCPC divides the semicircular area into two parts. A circle will be inscribed in each of them that touches AB,PCAB, PC and hh. The points of contact of the two circles with ABAB are denoted by DD and EE, where DD lies between AA and PP. Prove that the size of the angle DCEDCE does not depend on the choice of PP.
(Walther Janous)
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