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quadrilateral which is a cyclic quadrilateral and tangent qu

Source: Russian Olympiad 2004, problem 10.3

May 3, 2004
geometryincentertrigonometryratiocyclic quadrilateralexterior angleangle bisector

Problem Statement

Let ABCD ABCD be a quadrilateral which is a cyclic quadrilateral and a tangent quadrilateral simultaneously. (By a tangent quadrilateral, we mean a quadrilateral that has an incircle.) Let the incircle of the quadrilateral ABCD ABCD touch its sides AB AB, BC BC, CD CD, and DA DA in the points K K, L L, M M, and N N, respectively. The exterior angle bisectors of the angles DAB DAB and ABC ABC intersect each other at a point K′ K^{\prime}. The exterior angle bisectors of the angles ABC ABC and BCD BCD intersect each other at a point L′ L^{\prime}. The exterior angle bisectors of the angles BCD BCD and CDA CDA intersect each other at a point M′ M^{\prime}. The exterior angle bisectors of the angles CDA CDA and DAB DAB intersect each other at a point N′ N^{\prime}. Prove that the straight lines KK′ KK^{\prime}, LL′ LL^{\prime}, MM′ MM^{\prime}, and NN′ NN^{\prime} are concurrent.
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