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 Find GM, Get One Side? (Posted on 2007-06-11)
In triangle PQR; QR = p, PR = q and PQ = r with Angle QPR = 2* Angle PQR. The length of all the three sides of the triangle are different.

Is it always true that: p2 = q (q + r)?.

 See The Solution Submitted by K Sengupta Rating: 3.0000 (2 votes)

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 Solution | Comment 1 of 4
`From the law of sines,`
`    q          p           p              p -------- = -------- = --------- = ----------------  sin(Q)     sin(P)     sin(2Q)     2 sin(Q)cos(Q)`
`                      or             cos(Q) = p/2q               From the law of cosines,`
` q^2 = p^2 + r^2 - 2pr cos(Q)`
`     = p^2 + r^2 - 2pr (p/2q)`
`          or`
` p^2(q - r) = q(q + r)(q - r)`
`If q != r, then p^2 = q(q + r).`
`If q = r, then Q = R = 45 and P = 90`
`   and therefore p^2 = q^2 + r^2 = q^2 + qr`
`                     = q(q + r)`
`Thus, p^2 = q(q + r) whether q = r or not.`
` `

 Posted by Bractals on 2007-06-11 12:49:11

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