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 Find the Ratio (Posted on 2007-08-20)
I (radius=r) is the incircle of triangle ABC (right angled at B) and O (radius=R) is another circle which touches the extended side BC and AC. Find the ratio (R/r) if the point of intersection of circles I and O is midpoint of AC.

 Submitted by Praneeth Rating: 3.2500 (4 votes) Solution: (Hide) 3+2*√2 The tangents from an external point to the circle are equal. As Incircle touches hypotenuse at midpoint, ABC is isosceles right angledtriangle. => BC=AC/√2 Draw perpendiculars from I and O on to BC and points of intersection beP and Q respectively. Draw perpendicular from I to OQ to meet at R. http://perplexus.info/show.php?pid=5713&cid=38269 Now BC=r+PC, AC=2*PC. => (r+PC)*√2=2*PC => PC=(√2 +1)*r --(1) Consider triangle IOR IR=2*PC IO=r+R OR=R-r(Assume R>r) As it is aright angled triangle, 4*PC˛+(R-r)˛=(R+r)˛ => PC = √(R*r) Substitute this in eq(1) √(R*r)=(√2 +1)*r => √(R/r)=√2 +1 => R/r = 3+2*√2. Other solutions are given in comments by Bractals and Charlie.

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 Subject Author Date Answer K Sengupta 2008-08-10 06:34:25 solution Charlie 2007-08-20 11:48:05 Solution Bractals 2007-08-20 11:23:46
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