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IMO LongList 1992 SPA1

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September 2, 2010
geometryinradiusperimeterexradiusIMO ShortlistIMO Longlist

Problem Statement

A circle of radius ρ\rho is tangent to the sides ABAB and ACAC of the triangle ABCABC, and its center KK is at a distance pp from BCBC.
(a) Prove that a(p−ρ)=2s(r−ρ)a(p - \rho) = 2s(r - \rho), where rr is the inradius and 2s2s the perimeter of ABCABC. (b) Prove that if the circle intersect BCBC at DD and EE, then DE=4rr1(ρ−r)(r1−ρ)r1−rDE=\frac{4\sqrt{rr_1(\rho-r)(r_1-\rho)}}{r_1-r}
where r1r_1 is the exradius corresponding to the vertex A.A.
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