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### Area of Inscribed Circle

```
Date: 12/01/98 at 13:24:52
From: Anna
Subject: Trigonometry

The question is: Find the area of the circle inscribed in a triangle
a, b, c.  The hint is: Draw lines to the center of the circle and use
Heron's law.

I would appreciate it if you could help me!
Anna
```

```
Date: 12/01/98 at 15:41:37
From: Doctor Floor
Subject: Re: Trigonometry

Hi Anna,

Thanks for your question!

Let us consider triangle ABC with sidelengths BC = a, AC = b, AB = c
and center of the inscribed circle (incenter) I. Drop perpendicular
altitudes from I to Ia, Ib and Ic on the sides of ABC. The lengths of
these altitudes are equal to the radius r of the incircle. This radius
we will have to calculate in order to find the area of the inscribed
circle (= incircle).

C
/ \
/   \
/     \
Ib       Ia
/         \
/     I     \
/             \
A-------Ic------B

First, we know that area(ABC) = area(AIB) + area(BIC) + area(CIA).

We can calculate area(AIB) = 0.5 * AB * IIc = 0.5cr. And in the same
way we find area(BIC) = 0.5ar and area(CIA) = 0.5br.

The conclusion is that area(ABC) = 0.5(a+b+c)r = sr, where s is the
semiperimeter of ABC.

Then, from Heron's formula, we also know that:

area(ABC) = sqrt(s(s-a)(s-b)(s-c))

Combining the two formulas for the area of ABC we can derive:

sr = sqrt(s(s-a)(s-b)(s-c))
r = sqrt(s(s-a)(s-b)(s-c)) / s
r = sqrt((s-a)(s-b)(s-c) / s )

And now we can finish off the area of the incircle:

area(incircle) = pi * r^2
= pi * sqrt((s-a)(s-b)(s-c) / s )^2

pi(s-a)(s-b)(s-c)
= -----------------
s

Quite nice!

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
Middle School Conic Sections/Circles
Middle School Geometry
Middle School Triangles and Other Polygons

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