2020 CIME I Problems/Problem 11

Problem 11

An $excircle$ of a triangle is a circle tangent to one of the sides of the triangle and the extensions of the other two sides. Let $ABC$ be a triangle with $\angle ACB = 90$ and let $r_A, r_B, r_C$ denote the radii of the excircles opposite to $A, B, C$ , respectively. If $r_A=9$ and $r_B=11$ , then $r_C$ can be expressed in the form $m+\sqrt{n}$ , where $m$ and $n$ are positive integers and $n$ isn't divisible by the square of any prime. Find $m+n$ .

Solution

we know that ra*rb + rb*rc +rc*ra = s^2 therefore 11rc + 9rc + 99 = s^2 (eqn 1) also inradius = (a+b-c)/2 this implies that r+c=s. rc = ( area of triangle ABC )/s-c = (area of triangle ABC ) / r = s (eqn 2 ) then we get the equation s^2 - 20s -99 =0 (from eqn 1 and eqn 2 ) upon solving the quadratic we get s = 10 + root(199) hence rc = 10 +root(199) m+n = 209

         ~ Yash_R_Pillai

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2020 CIME I (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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2020 CIME I Problems/Problem 11

Problem 11

Solution

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