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Angle Bisectors (Posted on 2008-02-11) Difficulty: 3 of 5
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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re(2): Solution | Comment 3 of 5 |
(In reply to re: Solution by Dennis)

I said the converse of

"If AE = 2BD, then angle B is a right angle"

is easy to prove.

I also had a problem showing that

"If AE = 2BD, then angle B is a right angle"

But, since a 30-60-90 triangle meets the criteria of the problem, that's what I used.

I got

c^2 = a^2 - ab + b^2

and

b(b + c - a)(a + c)^2 = 4a(a + c - b)(b + c)^2

and gave up.

EDIT

After a lot of calculations using the above two equations (which I never want to do again), I was able to show that b = 2a and c = a*sqrt(3).

 Therefore, ABC is a 30-60-90 triangle.

Edited on February 11, 2008, 11:52 pm

Edited on February 12, 2008, 2:18 pm
  Posted by Bractals on 2008-02-11 20:47:24

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