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Source: Iranian TST 2021, second exam day 2, problem 2

May 22, 2021
geometrytrapezoidangle bisectorcircumcircle

Problem Statement

Point XX is chosen inside the non trapezoid quadrilateral ABCDABCD such that ∠AXD+∠BXC=180\angle AXD +\angle BXC=180. Suppose the angle bisector of ∠ABX\angle ABX meets the DD-altitude of triangle ADXADX in KK, and the angle bisector of ∠DCX\angle DCX meets the AA-altitude of triangle ADXADX in LL.We know BK⊥CXBK \perp CX and CL⊥BXCL \perp BX. If the circumcenter of ADXADX is on the line KLKL prove that KL⊥ADKL \perp AD.
Proposed by Alireza Dadgarnia
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