MathDB
similar quads wanted, 4 circumcenters related

Source: Polish MO Recond Round 1987 p6

September 9, 2024
geometrysimilaritysimilar

Problem Statement

We assign to any quadrilateral ABCD ABCD the centers of the circles circumscribed in the triangles BCD BCD , CDA CDA , DAB DAB , ABC ABC . Prove that if the vertices of a convex quadrilateral Q Q do not lie on the circle, then
a) the four points assigned to quadrilateral Q in the above manner are the vertices of the convex quadrilateral. Let us denote this quadrilateral by t(Q) t(Q) ,
b) the vertices of the quadrilateral t(Q) t(Q) do not lie on the circle,
c) quadrilaterals Q Q and t(t(Q) t(t(Q) are similar.