Christmas Streak 40/88: Circles, Lines and Points
Three circles with differing radii lie on a plane, and none of their centers are inside the other circles.
Now draw two common external tangents and draw their intersection for each pair of circles.
Is it necessarily true that the three points are collinear?
Details and Assumptions:
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We still consider the three points are collinear even if they coincide.
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A common external tangent is a common tangent that does not intersect the segment that joins the centers of the two circles.
This problem is a part of <Christmas Streak 2017> series .
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1 solution
This can be solved by vectors/geometrical methods, but here I'll show you the "genuine" method.
Think of three spheres placed on the ground. (= Plane G)
Then draw an escribed tangent cone for each pair of spheres. It's obvious that all of their vertices are on G.
Think of a plane that slices through all the three centers of the spheres. (= Plane P)
Since each vertex of the cones is on the line that connects the two spheres, the vertices must all be on P.
The vertices are on both P and G, and so we infer that they must be on the intersection of P and G, which is a straight line.
Then slice the whole diagram by plane P, and we see that this is what we were trying to prove. □