Circles Inscribed in TrianglesDate: 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/ |
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