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Show that the circumcircles of the triangles \(F R T, D P U\) and \(E Q S\) have a single point in common.


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A circle centered at the incenter \(I\) of a triangle \(A B C\) meets all three sides of the triangle: side \(B C\) at \(D\) and \(P\) (with \(D\) nearer to \(B\) ), side \(C A\) at \(E\) and \(Q\) (with \(E\) nearer to \(C\) ), and side \(A B\) at \(F\) and \(R\) (with \(F\) nearer to \(A\) ). The diagonals of the quadrilaterals \(E Q F R, F R D P\), and \(D P E Q\) meet at \(S, T\), and \(U\), respectively. Show that the circumcircles of the triangles \(F R T, D P U\) and \(E Q S\) have a single point in common.
in Mathematics by ♦ Platinum (97.7k points) | 128 views

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