See complete Problem 208 at:

gogeometry.com/problem/p208_triangle_excircles_angles.htm

Triangle, Excircles, Angles, 360 degrees. Level: High School, SAT Prep, College geometry.

## Sunday, November 16, 2008

### Elearn Geometry Problem 208: Triangle, Excircles, Angles, 360 degrees

Labels:
angle,
excircle,
tangency point,
triangle

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with the exteror angles of ABC ,and isoscele triangles like BEF we have:

ReplyDeletealpha = 180-(B/2+C/2)

beta = 180-(C/2+A/2)

gamma = 180-(A/2+B/2)

sum = 3.180-180=360

.-.

Let :

ReplyDelete∠ABC=s

∠ACB=u

∠BAC=t

Let O be the center of the circle E_a. Then join DO and OF.

Extend DO to meet the circle at the point P

∠DPF=180-α

therefore

∠DOF = 360-2α

DOFC lies on a circle therefore

(1.) 180+u=2α

Working in the same way we can find similar formules for β and θ

(2.)180+t=2β

(3.)180+s=2θ

Adding equations 1 ,2 and 3 with u+s+t=180

2(α+β+θ)=540+s+t+u

2(α+β+θ)=720

α+β+θ=360