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PEM Level C
9 January 2010
Topics in Geometry  Power of a Point and the Radical Axis
Theorems:
1.
Let
P
be any point, and
Σ
be a given circle. Let
‘
be a line through
P
that intersects
Σ
at
points
A
and
B
.
(a) If the
P
is inside
Σ
, then
P
(
P,
Σ) =

PA
·
PB
.
(b) If the
P
is outside
Σ
, then
P
(
P,
Σ) =
PA
·
PB
.
2. Let Σ
1
and Σ
2
be two nonconcentric circles. Then the locus of all points
P
such that
P
(
P,
Σ
1
) =
P
(
P,
Σ
2
) is a line.
*The line is called that radical axis of the nonconcentric circles
Σ
1
and
Σ
2
.
3. The radical axis of two circles that intersect at two distinct points is the line that passes
through the points of intersection of the circles. Also, the radical axis of two externally
tangent circles is the common internal tangent of the circles.
4. If the centers of three circles are noncollinear, then the radical axes of the pairwise circles
are concurrent.
Exercises:
1. Given three distinct points
A
,
B
, and
C
, prove that the locus of all points
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 Spring '09
 B
 Geometry

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