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 
radius of the large circle.

-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
Subject: Re: geometry question already addressed

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/