000RM.dvi

8.2 The circumcircle and the circumcircle 233

8.2 The circumcircle and the circumcircle

The perpendicular bisectors of the three sides of a triangle are concurrent
at thecircumcenterof the triangle. This is the center of the circumcircle,
the circle passing through the three vertices of the triangle.

O

C

A

B

D

F E O

C

A

B

D

Theorem 8.1(The law of sines). LetRdenote the circumradius of a
triangleABCwith sidesa,b,copposite to the anglesA,B,Crespec-
tively.
a
sinA

=

b

sinB

=

c

sinC

=2R.

Since the area of a triangle is given by =^12 bcsinA, the circumra-
dius can be written as

R=

abc

4

.