Section 4.5: Circles in Euclidean Geometry (Part 1: Through Corollary 4.5.13)
Definition:
Given a point P and a real number
r
>
0 ,
the
circle
C( P, r )
is the set
of points with a common distance
r
from
point P, i.e., the set of points Q in the
plane such that
PQ
=
r .
Point
P
is the
center
and
number
r
is the
radius
of the circle
C( P, r ) .
Theorem 4.5.1, The "Three Points Circle" Theorem:
In the Euclidean Plane, three distinct non-collinear points determine a unique circle.
Proof: Let A, B, and C be three non-collinear points. Let
l
and
m be the
perpendicular bisectors of
AB
and
BC
, resp.,
intersecting these segments at midpoints M and N,
resp. .
Suppose
l
is parallel to m.
By Theorem 3.4.8, if a
line is perpendicular to one of two parallel lines,
then it is perpendicular to the other.
Therefore,
since
AB
⊥
suuur
l
,
AB
⊥
suuur
m.
Let
D be the point where
AB
intersects m .
D is not equal to N or else A, B,
and C would be collinear. Thus,
∆
BDN has angle
sum greater than 180
°
, which is a impossible in
Euclidean Geometry.
Therefore,
l
is not parallel to
m.
Therefore,
l
and m
intersect, say a point P.
Since
P
is on the perpendicular bisectors of segments,
PA = PB
and
PB = PC by Theorem 3.2.8.
Let
r
=
PA
=
PB
=
PC . Then, all three points,
A, B, and C are on the circle
C( P, r ).
Any other
possible circle passing through A, B, and C must
have its center on both perpendicular bisectors and
so its center must be the point P of intersection of
these perpendicular bisectors, so there is no other
such circle than C( P, r ) passing through A, B, and
C.
Q E D
Note:
This proof provides a method for constructing the center of the circle.
l
m
D
M
N
C
B
A
l
m
P
M
N
A
B
C