The problem states that : "Given three objects in the plane, each of which may be a circle C, a point P (a degenerate circle), or a line L (part of a circle with infinite radius), find another circle that is tangent to (just touches) each of the three".
There are ten combinations possible through combinations of point, line and circle. The most difficult case, to find a tangent circle to any three other circles. The resultant solution is here.
These circles are known as Apollonius' Circles. A combination of 8 solutions.
If you observe carefully, the circles in the solution are generated in pairs and are conjugate to one another. Hence forming a symmetry.
Carrying forward this problem, if the 3 circles in the problem are mutually tangential to one another, then the 8 circles of the solution collapse into 2, as seen here
These circles are known as Soddy Circles.
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3 comments:
everything's going over my head!!!
Wasn't there a Apollonius theorem too??
Link alli iro ella articles odhadya?
Ya, there is an Apollonius' Theorem. It states that "Given a triangle ABC, if D is any point on BC such that it divides BC in the ratio n:m (or mBD = nDC), then
mAB2 + nAC2 = mBD2 + nDC2 + (m + n)AD2."
What you studied in Geometry was the simplest case of this theorem, i.e m=n. BD is the median of AC.
Nyapaka aaytha?
hoon aitu...
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