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Projections to $OI$

Source: 2023 Israel TST Test 7 P3

May 9, 2023
TSTgeometrytriangle centerscircumcircleincenter

Problem Statement

Let ABCABC be an acute-angled triangle with circumcenter OO and incenter II. The midpoint of arc BCBC of the circumcircle of ABCABC not containing AA is denoted SS. Points E,FE, F were chosen on line OIOI for which BEBE and CFCF are both perpendicular to OIOI. Point XX was chosen so that XE⊥ACXE\perp AC and XF⊥ABXF\perp AB. Point YY was chosen so that YE⊥SCYE\perp SC and YF⊥SBYF\perp SB. DD was chosen on BCBC so that DI⊥BCDI\perp BC. Prove that XX, YY, and DD are collinear.
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