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G3

Part of 2017-IMOC

Problems(1)

IMOC 2017 G3 (AB CX DX)^2+(CD AX BX)^2=(AD BX CX)^2 +(BC AX DX)^2

Source: https://artofproblemsolving.com/community/c6h1740077p11309077

3/20/2020
Let ABCDABCD be a circumscribed quadrilateral with center OO. Assume the incenters of △AOC,△BOD\vartriangle AOC, \vartriangle BOD are I1,I2I_1, I_2, respectively. If circumcircles of △AI1C\vartriangle AI_1C and △BI2D\vartriangle BI_2D intersect at XX, prove the following identity: (AB⋅CX⋅DX)2+(CD⋅AX⋅BX)2=(AD⋅BX⋅CX)2+(BC⋅AX⋅DX)2(AB \cdot CX \cdot DX)^2 + (CD\cdot AX \cdot BX)^2 = (AD\cdot BX \cdot CX)^2 + (BC \cdot AX \cdot DX)^2
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