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 Angle Bisectors (Posted on 2008-02-11)
In triangle ABC, angle C has a measure of 60 degrees. Point D lies on side AC so that BD bisects angle B and BD = 1. Similarly, point E lies on side BC so that AE bisects angle A and AE = 2.

Find the area of triangle ABC.

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

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 AE = 2BD implies B = 90 degrees | Comment 4 of 5 |
`OOPS - Should not of put the hint in the title.`
`The length of the medians are`
`                      a   |AE|^2 = bc[1 - (-----)^2]                     b+c`
`        and`
`                      b   |BD|^2 = ac[1 - (-----)^2]                     a+c`
`Therefore, |AE| = 2|BD| implies`
`   b(b+c-a)(a+c)^2 = 4a(a+c-b)(b+c)^2    (1)`
`From the Law of Cosines,`
`   c^2 = a^2+b^2-2ab*cos(C)`
`       or`
`   c^2 = a^2+b^2-ab                      (2)`
`Equations (1) and (2) imply (with a lot of algebra)`
`   (4a^2-b^2)[c(a+b)+(a^2+b^2)] = 0`
`This implies`
`   b = 2a`
`this with equation (2) gives `
`   c = 3*sqrt(3)`
`Therefore,`
`   b^2 = (2a)^2 = a^2+(a*sqrt(3))^2 `
`       = a^2+c^2`
`Therefore,`
`   B = 90 degrees`
` `

Edited on February 12, 2008, 4:13 pm

Edited on February 12, 2008, 4:15 pm

Edited on February 12, 2008, 11:53 pm
 Posted by Bractals on 2008-02-12 16:11:40

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