MathDB
Scalene triangles with circles

Source: 2014 BAMO-8 #4, 2014 BAMO-12 #2

February 22, 2016
geometry

Problem Statement

Let △ABC\triangle{ABC} be a scalene triangle with the longest side ACAC. (A <spanclass=′latex−italic′>scalenetriangle</span>{<span class='latex-italic'>scalene triangle</span>} has sides of different lengths.) Let PP and QQ be the points on the side ACAC such that AP=ABAP=AB and CQ=CBCQ=CB. Thus we have a new triangle △BPQ\triangle{BPQ} inside △ABC\triangle{ABC}. Let k1k_1 be the circle circumscribed around the triangle △BPQ\triangle{BPQ} (that is, the circle passing through the vertices B,P,B,P, and QQ of the triangle △BPQ\triangle{BPQ}); and let k2k_2 be the circle inscribed in triangle △ABC\triangle{ABC} (that is, the circle inside triangle △ABC\triangle{ABC} that is tangent to the three sides AB,BCAB,BC, and CACA). Prove that the two circles k1k_1 and k2k_2 are concentric, that is, they have the same center.
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