Ratios Involving Six Radii

Given a point on the side of the triangle , let and be the radii of the inscribed circles of the triangles and , and let and be the radii of the circumcircles of the triangles and . Let and denote the radii of the inscribed circle and the circumcircle of the triangle , respectively. Prove that .

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The label for each radius is placed at the center of the circle of which it is the radius.
This Demonstration comes from problem 8 of the shortlisted problems for the 1970 International Mathematical Olympiad (IMO).
Reference
[1] D. Djukić, V. Janković, I. Matić, and N. Petrović, The IMO Compendium, 2nd ed., New York: Springer, 2011 p. 69.
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