See complete Problem 206 at:

gogeometry.com/problem/p206_right_triangle_area_inradius.htm

Area of a Right Triangle Area, Inradius, Exradius relative to the hypotenuse. Level: High School, SAT Prep, College geometry

Post your solutions or ideas in the comments.

## Saturday, November 15, 2008

### Elearn Geometry Problem 206: Area of a Right Triangle, Inradius, Exradius

Labels:
area,
exradius,
hypotenuse,
inradius,
right triangle

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Solution of problem 206.

ReplyDeleteLet s be the semi perimeter of triangle ABC. The area of this triangle is equal to S = s.r. By problem 201, we have ra = s. So S = r.ra.

Let AB=a, AC=b and area right triangle ABC=s

ReplyDeleteBy Poncelet we have: a+b=c+2r

(ra-r)+(ra-r)=a+b-2r

Simplifying

ra+r=a+b

Now we know that

(ab)/2=(a-r)(b-r)

s=(a-r)(b-r)

Rewriting this expression

r^2=r(a+b)-s................(1)

ra+r=a+b can be rewrite this way: rar+r^2=r(a+b)

rar+r^2=r(a+b)...............(2)

Now

rar+r(a+b)-s=r(a+b)

s=rar

Q.E.D.

By Tony Garcia