Associated Topics || Dr. Math Home || Search Dr. Math

### Circles Inscribed in Triangles

```
Date: 11/14/96 at 10:05:42
From: Omer Moltay
Subject: Geometry problem involving circles inscribed in triangles.

A triangle ABC has the following characteristics: A circle with
radius r is inscribed in it. Three new triangles (the chopped off
corners of the triangle ABC) are formed in triangle ABC by drawing
segments parallel to the sides of triangle ABC. Circles are inscribed
in these three new triangles, with radii r1,r2 and r3.

Prove that r1 + r2 + r3 = r.
```

```
Date: 01/11/97 at 16:23:51
From: Doctor Donald
Subject: Re: Geometry problem involving circles inscribed in
triangles.

A nice way to solve this problem is to extend the line segments which
are parallel to the three sides of the triangle and cut off the small
triangles in the corners, so that the three lines form a large
triangle.  Observe that this large triangle is congruent to the
original large triangle, with the same inscribed circle.

Look at any corner of the original triangle, and note that the small
triangle in that corner has one of its sides lying along a side of the
new large triangle. Also along that side of the new triangle are the
triangles in the two adjacent corners of the new triangle.

All of these corner triangles are similar to the large triangles and
to each other. The three corner triangles I have selected represent
all three corners of the original triangle, and we can see that the
sum of the lengths of the corresponding sides of the corner triangles
is exactly the length of that side of the new large triangle.

Since the sum of the lengths of corresponding sides of the corner
triangles equals the length of the corresponding side of the large
triangle, the same is true of any corresponding components, in
particular that the radii of the small inscribed circles add up to the

-Doctor Donald, The Math Forum
Check out our Web site  http://mathforum.org/dr.math/
```

```
Date: 06/11/2001 at 13:58:49
From: Tanya Mills
Subject: Re: Geometry problem involving circles inscribed in
triangles.

Are you assuming that the triangle is equalateral and that the three
triangles in the corners of the triangle are tangent on one side to
the first inscribed circle?
```

```
Date: 06/11/2001 at 16:12:16
From: Doctor Rick

Hi, Tanya.

No and yes.

It is not assumed that the triangle is equilateral (though the figure
appears equilateral). Here is a figure of the problem, showing a clearly
non-equilateral triangle:

The lines that cut off the corner triangles are parallel to the opposite
sides of the original triangle, and tangent to the circle inscribed in the
original triangle. This is not stated very clearly in the question.

The proof omits a critical point: not only are all the small triangles
similar to the original triangle, but each is *congruent* to the opposite
triangle. To see this, draw lines from the incenter D to each intersection
between a side of the triangle and a line parallel to another side, and also
from D to each point of tangency between the inscribed circle and the
original triangle. You will create 12 triangles. Each of these is congruent
to the opposite triangle, because respective sides are parallel and a side
of each is the radius of the inscribed circle. Thus each side of the hexagon
formed by the 12 triangles is congruent to the opposite side.

Armed with this fact, we can look at the bottm side of the original triangle
and see that it is the sum of the corresponding sides of two of the cut-off
triangles, plus a triangle that is congruent to the third. Thus the radii of
the circles inscribed in the three small triangles sum to the radius of the
circle inscribed in the original triangle.

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

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search