MathDB
Quadrilateral geo with incircles

Source: KoMaL A862

November 11, 2023
geometrykomal

Problem Statement

Let ABCDABCD be a cyclic quadrilateral inscribed in circle ω\omega. Let FA,FB,FCF_A, F_B, F_C and FDF_D be the midpoints of arcs AB,BC,CDAB, BC, CD and DADA of ω\omega. Let IA,IB,ICI_A, I_B, I_C and IDI_D be the incenters of triangles DAB,ABC,BCDDAB, ABC, BCD and CDACDA, respectively.
Let ωA\omega_A denote the circle that is tangent to ω\omega at FAF_A and also tangent to line segment CDCD. Similarly, let ωC\omega_C denote the circle that is tangent to ω\omega at FCF_C and tangent to line segment ABAB.
Finally, let TBT_B denote the second intersection of ω\omega and circle FBIBICF_BI_BI_C different from FBF_B, and let TDT_D denote the second intersection of ω\omega and circle FDIDIAF_DI_DI_A.
Prove that the radical axis of circles ωA\omega_A and ωC\omega_C passes through points TBT_B and TDT_D.
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