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Finally an actual geometry problem!

Source: 2023 Olympic Revenge problem 5

March 28, 2023
geometrycircumscribed quadrilateralIncenters

Problem Statement

Let ABCDABCD be a circumscribed quadrilateral and T=AC∩BDT=AC\cap BD. Let I1I_1, I2I_2, I3I_3, I4I_4 the incenters of ΔTAB\Delta TAB, ΔTBC\Delta TBC, TCDTCD, TDATDA, respectively, and J1J_1, J2J_2, J3J_3, J4J_4 the incenters of ΔABC\Delta ABC, ΔBCD\Delta BCD, ΔCDA\Delta CDA, ΔDAB\Delta DAB. Show that I1I2I3I4I_1I_2I_3I_4 is a cyclic quadrilateral and its center is J1J3∩J2J4J_1J_3\cap J_2J_4
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