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### Radius of a Circle Inscribed in a Triangle

```
Date: 06/02/99 at 19:51:38
From: Michael
Subject: Radius of an inscribed circle

There is a drawing of a right triangle with sides of lengths 3, 4, and
5. A circle is inscribed in the triangle so that each side is tangent
to the circle. The question is: What is the radius of an inscribed
circle of a triangle with sides 3, 4, and 5?

I have tried a few different methods, none of which seems to work.
What should I do?
```

```
Date: 06/03/99 at 08:03:23
From: Doctor Floor
Subject: Re: Radius of an inscribed circle

Hi, Michael,

Let's consider the general situation of a right triangle:

B
|  \
|     Z            I is the center of the inscribed circle.
X   I    \         X,Y and Z are the points where the inscribed
|           \      circle meets the sides, so XI = YI = ZI = r.
C---Y----------A

I write a = BC (length), b = AC and c = AB.

We know that area(ABC) = 0.5*a*b   .........................[1].

Also area(ABC) = area(ABI) + area(CIA) + area(BCI)
= 0.5*r*a + 0.5*r*b + 0.5*r*c
= 0.5*r*(a+b+c)      .........................[2].

From [1] = [2] we find the formula:

r = a*b/(a+b+c).

I suppose you can do the rest yourself now. I hope this helps you out.

Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School Triangles and Other Polygons

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