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Areas and circumcenters

Source: Kvant Magazine No. 11-12 2022 M2727

March 7, 2023
Kvantgeometryareas

Problem Statement

A convex quadrilateral ABCDABCD is given. Let OaO_a be the circumcenter of the triangle DBCDBC, and define Ob,OcO_b,O_c and OdO_d similarly. The points Oa,Ob,Oc,OdO_a, O_b, O_c, O_d are the vertices of a convex quadrilateral. Prove that its area is equal to half of the absolute value of the difference between the areas of AObCOdAO_bCO_d and BOcDOaBO_cDO_a.
Proposed by V. Dubrovsky
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