MathDB
locus and similarity

Source: Indonesia IMO 2007 TST, Stage 2, Test 2, Problem 1

November 15, 2009
geometrycircumcircleincentergeometry proposed

Problem Statement

Given triangle ABC ABC and its circumcircle Γ \Gamma, let M M and N N be the midpoints of arcs BC BC (that does not contain A A) and CA CA (that does not contain B B), repsectively. Let X X be a variable point on arc AB AB that does not contain C C. Let O1 O_1 and O2 O_2 be the incenter of triangle XAC XAC and XBC XBC, respectively. Let the circumcircle of triangle XO1O2 XO_1O_2 meets Γ \Gamma at Q Q. (a) Prove that QNO1 QNO_1 and QMO2 QMO_2 are similar. (b) Find the locus of Q Q as X X varies.
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