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intersect on the incircle

Source: Serbian Mathematical Olympiad 2007

June 9, 2007
geometryprojective geometrycomplex numberscyclic quadrilateralgeometry proposed

Problem Statement

In a scalene triangle ABC,AD,BE,CFABC , AD, BE , CF are the angle bisectors (D∈BC,E∈AC,F∈AB)(D \in BC , E \in AC , F \in AB). Points Ka,Kb,KcK_{a}, K_{b}, K_{c} on the incircle of triangle ABCABC are such that DKa,EKb,FKcDK_{a}, EK_{b}, FK_{c} are tangent to the incircle and Ka∉BC,Kb∉AC,Kc∉ABK_{a}\not\in BC , K_{b}\not\in AC , K_{c}\not\in AB. Let A1,B1,C1A_{1}, B_{1}, C_{1} be the midpoints of sides BC,CA,ABBC , CA, AB , respectively. Prove that the lines A1Ka,B1Kb,C1KcA_{1}K_{a}, B_{1}K_{b}, C_{1}K_{c} intersect on the incircle of triangle ABCABC.
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