Angle bisectors of trianglesIn triangle ABC, angle bisectors A D ¯ ,, B E ¯ ,, and C F ¯ meet at I. If D I = 3 ,, B D = 4 ,, and B I = 5 ,, then compute the area of triangle ABC.

Question

Angle bisectors of trianglesIn triangle ABC, angle bisectors             A      D        ¯    ,,             B      E        ¯    ,, and             C      F        ¯   meet at I. If   D  I  =  3  ,,   B  D  =  4  ,, and   B  I  =  5  ,, then compute the area of triangle ABC.

Melany Flynn · Accepted Answer

Step 1Without using trig.From the fact that   △  B  D  I has side lengths 3,4,5 and the Pythagorean theorem, we know that angle D is a right triangle.If D is a right angle,   △  A  B  D is isosceles.This figure is only half of ABC...There is a theorem regarding bisected angles that says that if IB bisects B then   A  I  :  A  B as   D  I  :  D  B.  A  I  :  A  B  =  3  :  4Step 2 Let   A  I  =  3  x  ,  A  B  =  4  xFrom Pythagoras:   B      D    2    +  (  D  A      )    2    =  A      B    2          4    2    +  (  3  +  3  x      )    2    =  (  4  x      )    2  Now for some algebra:  16  +  9  +  18  x  +  9      x    2    =  16      x    2      7      x    2    −  18  x  −  25  =  0    (  7  x  −  25  )  (  x  +  1  )We can reject,   x  =  −  1 as lengths must be greater than 0.  A  I  =      75    7      A  D  =      75    7    +  3  =      96    7    A  r  e  a  =      1    2    (  A  C  )  (  A  D  )  =  4      96    7    =      384    7

Gauge Roach · Accepted Answer

Step 1if   ∠  B  D  I is a right angle then so is   ∠  C  D  I so you have symmetry across AD and an isosceles triangle - which helps a lot. You also have DI as the radius of the in-circle, and if you join I to the other two points G and H where the edges touch the in-circle then you do get similar triangles   △  A  D  B and   △  A  G  I, as well as congruent triangles   △  I  D  B and   △  I  G  B.Step 2So                     A        G            3        =                    A        D            4       and   (  A  G  +  4      )    2    =  A      D    2    +      4    2  which you can solve to give   A  D  =      96    7  and so the area is   2  ×      1    2    ×      96    7    ×  4  =      384    7    ≈  54.857.

In triangle ABC, angle bisectors bar{AD}, bar{BE}, and bar{CF} meet at I. If DI=3, BD=4, and BI=5, then compute the area of triangle ABC.

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