tangent circles

Draw  perpendiculars from the centers of  both  circles  to  a  side of the  triangle

Using similar triangles  we  have  that

(h - 5)  / 1  =  ( h - 2) /2

2 (h - 5) = 1 ( h -2)

2h  -10 =   h - 2

h = 8

Let the center of the top circle =  D   the apex  = A   and the  point where the radius of this circle meets the  side of the triangle, E....angle ACB   =  90.....and   AD = 3    DE  = 1

So

AE = sqrt ( AD^2 + DE^2)  =  sqrt( 3^2- 1^2)  = sqrt(8)

tan  DAE  =  DE/  AE   =  1/sqrt (8)

So

tan DAE  =  (base/2) / height

1/sqrt (8)   =base/2 * 8

8/sqrt (8)   =  base/2  =sqrt (8)

So   [ ABC ]    =   ( base / 2)  * height   =  8 sqrt (8)

Circles with radii 1 and 2 are inscribed in triangle ABC, as shown.  Find the area of triangle ABC.

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XY = 3       ZY = 1        WY = 2

∠BAW = sin^-1(1/3)

∠ABW = 90 - ∠BAW              ∠YBW = 1/2 (∠ABW)

BW = YW / tan(∠YBW)            AW = BW * tan(ABW)

[ABC] = AW * BW