We have three circles w1, w2 and Γ at the same side of line l such that w1 and w2 are tangent to l at K and L and to Γ at M and N, respectively. We know that w1 and w2 do not intersect and they are not in the same size. A circle passing through K and L intersect Γ at A and B. Let R and S be the reflections of M and N with respect to l. Prove that A,B,R,S are concyclic.
Problem
Source: Turkey TST 2022 P4 Day 2
Tags: geometry, geometry proposed, circles, tangent circles
electrovector
13.03.2022 14:39
Let O1,O,O2 be the centers of w1,w2 and Γ. ∠MKL=∠KO1M2=360−∠MON−∠NO2L2=∠ONM+∠O2NL which proves that KMNL is concyclic. Also we have that ABLK and MNBA are concyclic which means AB, KL and MN intersects at some point, T. It is also easy to see that MN, RS and KL intersect at some point which means T∈RS. Now we have TR⋅TS=TM⋅TN=TA⋅TB which means ABSR is concyclic.
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badumtsss
20.03.2022 17:28
Because of my amazing geometry skills, I couldn't see the direct angle chasing. Anyway, by Monge Theorem, we know that the center of homotety sending ω1 to ω2 is the intersection of KL and MN. Take the second intersections X, Y of MN and ω1, ω2, angle chasing shows that KLXY is cyclic, which also shows that KLMN is cyclic. MN and RS intersect on KL which ends the proof.