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PQ/CQ<= [AB/(AC+CB)]^2 , angle bisector and circumcircle related

Source: Germany Federal - Bundeswettbewerb Mathematik 2009, round 2, p3

April 9, 2020
geometryangle bisectorcircumcirclegeometric inequality

Problem Statement

Given a triangle ABCABC and a point PP on the side ABAB . Let QQ be the intersection of the straight line CPCP (different from CC) with the circumcicle of the triangle. Prove the inequality PQ‾CQ‾≤(AB‾AC‾+CB‾)2\frac{\overline{PQ}}{\overline{CQ}} \le \left(\frac{\overline{AB}}{\overline{AC}+\overline{CB}}\right)^2 and that equality holds if and only if the CPCP is bisector of the angle ACBACB. https://cdn.artofproblemsolving.com/attachments/b/1/068fafd5564e77930160115a1cd409c4fdbf61.png
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