MathDB

2

Part of 2022 Korea National Olympiad

Problems(1)

Parallel Lines of Circumcenters

Source: KMO 2022 P2

10/29/2022
In a scalene triangle ABCABC, let the angle bisector of AA meets side BCBC at DD. Let E,FE, F be the circumcenter of the triangles ABDABD and ADCADC, respectively. Suppose that the circumcircles of the triangles BDEBDE and DCFDCF intersect at P(≠D)P(\neq D), and denote by O,X,YO, X, Y the circumcenters of the triangles ABC,BDE,DCFABC, BDE, DCF, respectively. Prove that OPOP and XYXY are parallel.
geometrycircumcircleCircumcenterparallel