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Part of 2017 Kazakhstan National Olympiad

Problems(1)

Kazakhstan National Olympiad 2017, Final Round, 10-Grade, P4

Source: Kazakhstan National Olympiad 2017, Final Round, 10-Grade, P4

3/18/2017
The acute triangle ABCABC (AC>BC)(AC> BC) is inscribed in a circle with the center at the point OO, and CDCD is the diameter of this circle. The point KK is on the continuation of the ray DADA beyond the point AA. And the point LL is on the segment BDBD (DL>LB)(DL> LB) so that ∠OKD=∠BAC\angle OKD = \angle BAC, ∠OLD=∠ABC\angle OLD = \angle ABC. Prove that the line KLKL passes through the midpoint of the segment ABAB.
geometry