

Problem 548: Triangle, Transversal, Complete Quadrilateral,
Circumcircles, Circumcenters, Similarity, Concyclic
Points. Level: High School, SAT Prep, College
Geometry

The figure shows a triangle ABC with a
transversal DEF (F on the extension of AC). O_{1}, O_{2},
O_{3}, and O_{4} are the circumcenters of triangles
ABC, BDE, ECF, and ADF, respectively. Prove that (1) the circumcircles
of triangles ABC, BDE, ECF, and ADF meet at a point G, (2) Triangles ABC
and O_{4}O_{2}O_{3}. are similar, (3) O_{1},
O_{2}, O_{3}, O_{4}, and G are concyclic points
(lie on a same circle).
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