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Circumcenters form a circumscribed quadrilateral.

Source: Israel Olympic Revenge 2019 P3, also SL G7.

May 4, 2022
geometrycircumcircleangle bisectorincenter

Problem Statement

Let ABCDABCD be a circumscribed quadrilateral, assume ABCDABCD is not a kite. Denote the circumcenters of triangle ABC,BCD,CDA,DABABC,BCD,CDA,DAB by OD,OA,OB,OCO_D,O_A,O_B,O_C respectively. a. Prove that OAOBOCODO_AO_BO_CO_D is circumscribed. b. Let the angle bisector of ∠BAD\angle BAD intersect the angle bisector of ∠OBOAOD\angle O_BO_AO_D in XX. Similarly define the points Y,Z,WY,Z,W. Denote the incenters of ABCD,OAOBOCODABCD, O_AO_BO_CO_D by I,JI,J respectively. Express the angles ∠ZYJ,∠XYI\angle ZYJ,\angle XYI in terms of angles of quadrilateral ABCDABCD.
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