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Geometry is objectively cooler when "it turns out that"

Source: RMM Extralist 2021 G2

September 18, 2023
geometryRMM ShortlistInversionsimilaritiesBashingincentercircumcircle

Problem Statement

Let ABCABC be a triangle with incenter II. The line through II, perpendicular to AIAI, intersects the circumcircle of ABCABC at points PP and QQ. It turns out there exists a point TT on the side BCBC such that AB+BT=AC+CTAB + BT = AC + CT and AT2=ABā‹…ACAT^2 = AB \cdot AC. Determine all possible values of the ratio IP/IQIP/IQ.
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