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Geometry Mathley 7.3 tangential wanted and given

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June 7, 2020
geometrytangentialincircleTangents

Problem Statement

Let ABCDABCD be a tangential quadrilateral. Let ABAB meet CDCD at E,ADE, AD intersect BCBC at FF. Two arbitrary lines through EE meet AD,BCAD,BC at M,N,P,QM,N, P,Q respectively (M,N∈ADM,N \in AD, P,Q∈BCP,Q \in BC). Another arbitrary pair of lines through FF intersect AB,CDAB,CD at X,Y,Z,TX, Y,Z, T respectively (X,Y∈ABX, Y \in AB,Z,T∈CDZ, T \in CD). Suppose that d1,d2d_1, d_2 are the second tangents from EE to the incircles of triangles FXY,FZT,d3,d4FXY, FZT,d_3, d_4 are the second tangents from FF to the incircles of triangles EMN,EPQEMN,EPQ. Prove that the four lines d1,d2,d3,d4d_1, d_2, d_3, d_4 meet each other at four points and these intersections make a tangential quadrilateral.
Nguyễn Văn Linh
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