First some definitions.
The power of a point P with respect to a circle C with center O and radius r
is defined by ℘(P,C) = |OP|2 - r2. Clearly it is negative, zero, or positive
depending on whether P is inside, on, or outside circle C.
The radical axis (or radical line) with respect to two nonconcentric circles
is
defined as the locus of points whose power respect to both circles is the same.
It is easy to show that it is a line perpendicular to the line joining the centers
of the circles. If the circles intersect in two points, then the radical axis is the
secant defined by the two points.
In our problem let C1, C2, and C3 be the the three circles and S12, S23, and
S31 be the common secants (with subscripts denoting which circles). Let S12
and S23 intersect at point P.
P∈S12 ⇒ ℘(P,C1) = ℘(P,C2)
P∈S23 ⇒ ℘(P,C2) = ℘(P,C3)
⇒ ℘(P,C1) = ℘(P,C3) ⇒ P∈S31
Therefore, the secants are concurrent.
QED
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