MathDB
2011 BAMO 6 |AO|/|AA'|+ |BO|/|BB'|+|CO|/|CC'|= 1, 3 circle related

Source:

August 27, 2019
geometrycirclesratio

Problem Statement

Three circles k1,k2k_1, k_2, and k3k_3 intersect in point OO. Let A,BA, B, and CC be the second intersection points (other than OO) of k2k_2 and k3,k1k_3, k_1 and k3k_3, and k1k_1 and k2k_2, respectively. Assume that OO lies inside of the triangle ABCABC. Let lines AO,BOAO,BO, and COCO intersect circles k1,k2k_1, k_2, and k3k_3 for a second time at points A′,B′A', B', and C′C', respectively. If ∣XY∣|XY| denotes the length of segment XYXY, prove that ∣AO∣∣AA′∣+∣BO∣∣BB′∣+∣CO∣∣CC′∣=1\frac{|AO|}{|AA'|}+\frac{|BO|}{|BB'|}+\frac{|CO|}{|CC'|}= 1
Back to ProblemsView on AoPS