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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.

 See The Solution Submitted by Praneeth Rating: 3.2500 (4 votes)

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 Solution | Comment 1 of 3
`If the incircle of triangle ABC touches themidpoint of the hypotenuse, then ABC is aisoceles right triangle.`
`Let L and M be the points where the incircletouches sides AB and AC rspectively.`
`       |AC|^2 = |AB|^2 + |BC|^2`
`    (2|AM|)^2 = 2|AB|^2`
`      4|AM|^2 = 2(|AL| + |LB|)^2`
`              = 2(|AM| + r)^2`
`             or`
`            0 = 2|AM|^2 - 4r|AM| - 2r^2`
`             or`
`         |AM| = r(1 + sqrt(2))`
`From similar triangles OMA and AMC we get`
`        |OM|     |AM|       ------ = ------        |AM|     |IM|`
`             or`
`           rR = |AM|^2`
`             or`
`           R          --- = 3 + 2sqrt(2) ~= 5.828427           r         `
` `

 Posted by Bractals on 2007-08-20 11:23:46

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