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 Incircle Points (Posted on 2015-11-04)
Consider a triangle PQR where PQ = 2014, QR = 2015, RP =2016. RM is an altitude of this triangle.

Determine the distance between the points at which the incircles of PRM and QRM touch RM.

 No Solution Yet Submitted by K Sengupta No Rating

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 Solution Comment 1 of 1
`Let a = PM, r-a = MQ, h = RM, p = QR, and q = RP.`
`Applying the Pythagorean theorem to trianglesPMR and QMR gives`
`   h^2 = q^2 - a^2 = p^2 - (r-a)^2              ==>   2a = (q^2 + r^2 - p^2)/r`
`The distance between tangency points is`
`       |  a + h - q     (r-a) + h - p  |   d = | ----------- - --------------- |       |      2                2       |`
`       |  p - q - r + 2a  |     = | ---------------- |       |         2        |       |  p - q - r + (q^2 + r^2 - p^2)/r  |     = | --------------------------------- |       |                  2                |`
`     = | q(q - r) - p(p - r) |/(2r)`
`For our problem,`
`   d = | 2016(2016 - 2014) - 2015(2015 - 2014) | / 4028`
`     = 2017/4028 ~= 0.5007447864945382323733862959285`
`QED`

Edited on November 4, 2015, 2:07 pm
 Posted by Bractals on 2015-11-04 14:05:04

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