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|PM-PN|/PC is constant , MC _|_ NC, semicircle of diameter AB and center C

Source: 2011 Saudi Arabia Pre-TST February 4.1 https://artofproblemsolving.com/community/c2745403_2011_

January 1, 2022
ratiofixedgeometrysemicircle

Problem Statement

On a semicircle of diameter ABAB and center CC, consider vari­able points MM and NN such that MC⊥NCMC \perp NC. The circumcircle of triangle MNCMNC intersects ABAB for the second time at PP. Prove that ∣PM−PN∣PC\frac{|PM-PN|}{PC} constant and find its value.
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