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locus is a line, point in circle fixed (2016 Kyiv City MO 10.4)

Source:

September 6, 2020
geometryLocuscirclecollinear

Problem Statement

On the circle with diameter ABAB, the point MM was selected and fixed. Then the point Qi{{Q} _ {i}} is selected, for which the chord MQiM {{Q} _ {i}} intersects ABAB at the point Ki{{K} _ {i}} and thus ∠MKiB<90∘ \angle M {{K} _ {i}} B <90 {} ^ \circ. A chord that is perpendicular to ABAB and passes through the point Ki{{K} _ {i}} intersects the line BQiB {{Q} _ {i}} at the point Pi{{P } _ {i}}. Prove that the points Pi{{P} _ {i}} in all possible choices of the point Qi{{Q} _ {i}} lie on the same line.
(Igor Nagel)
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