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Unformatted text preview: Circumscribed and Inscribed Circles • Section 2.5 63 We will use Figure 2.5.6 to ﬁnd the radius r of the inscribed circle. Since O A bisects A ,
r
we see that tan 1 A = AD , and so r = AD · tan 1 A . Now, △ O AD and △ O AF are equivalent
2
2
triangles, so AD = AF . Similarly, DB = EB and FC = CE . Thus, if we let s = 1 (a + b + c), we
2
see that
2 s = a + b + c = ( AD + DB) + (CE + EB) + ( AF + FC ) = AD + EB + CE + EB + AD + CE = 2 ( AD + EB + CE ) s = AD + EB + CE = AD + a AD = s − a .
Hence, r = ( s − a) tan 1
2 A. Similar arguments for the angles B and C give us: 1
Theorem 2.10. For any triangle △ ABC , let s = 2 (a + b + c). Then the radius r of its inscribed
circle is
1
1
(2.38)
r = ( s − a) tan 2 A = ( s − b) tan 2 B = ( s − c) tan 1 C .
2 We also see from Figure 2.5.6 that the area of the triangle △ AOB is
Area(△ AOB) = 1
2 base × height 1
2 = cr . 1
Similarly, Area(△ BOC ) = 1 a r and Area(△ AOC ) = 2 b r . Thus, the area K of △ ABC is
2 K = Area(△ AOB) + Area(△ BOC ) + Area(△ AOC ) =
= r= 1
2 (a + b + c) r K
=
s 1
2 cr + 1
2 ar + 1
2 br = sr , so by Heron’s formula we get s ( s − a) ( s − b ) ( s − c )
=
s s ( s − a) ( s − b ) ( s − c )
=
s2 ( s − a) ( s − b ) ( s − c )
.
s We have thus proved the following theorem:
1
Theorem 2.11. For any triangle △ ABC , let s = 2 (a + b + c). Then the radius r of its inscribed
circle is
K
( s − a) ( s − b ) ( s − c )
=
.
(2.39)
r=
s
s Recall from geometry how to bisect an angle: use a compass centered at the vertex to draw an arc that intersects the sides of the
angle at two points. At those two points use a compass to draw an
arc with the same radius, large enough so that the two arcs intersect at a point, as in Figure 2.5.7. The line through that point and
the vertex is the bisector of the angle. For the inscribed circle of a
triangle, you need only two angle bisectors; their intersection will be
the center of the circle. d
d
A
Figure 2.5.7 ...
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This note was uploaded on 01/21/2012 for the course MAC 1130 taught by Professor Dr.cheun during the Fall '11 term at FSU.
 Fall '11
 Dr.Cheun
 Calculus, Angles

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