IMO 2006 Shortlist G7
In a triangle ABC, let Ma,Mb,Mc be respectively the midpoints of the sides BC, CA, AB and Ta,Tb,Tc be the midpoints of t...
Category: Geometry
Problem
In a triangle ABC, let Ma,Mb,Mc be respectively the midpoints of the sides BC, CA, AB and Ta,Tb,Tc be the midpoints of the arcs BC, CA, AB of the circumcircle of ABC, not containing the opposite vertices. For i ∈ {a,b,c}, let ωi be the circle with MiTi as diameter. Let pi be the common external tangent to ωj, ωk ({i,j,k} = {a,b,c}) such that ωi lies on the opposite side of pi than ωj, ωk do. Prove that the lines pa,pb,pc form a triangle similar to ABC and find the ratio of similitude. (Slovakia)