Three circle's tangent line
Draw three circles with distinct radii on a 2D plane.
As shown, for each pair of the three circles, draw two tangent lines and get their intersection point.
Are the 3 intersection points on the same line (as illustrated in blue)?
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1 solution
First, imagine that we have 3 spheres on a plane that have the same radii as the circles. Then we have a 2nd plane that is tangent to all 3, which intersects the 1st plane in a line. Then the 3 cones that are tangent to each pair of spheres are also tangent to both planes, and their apexes are colinear. That means there's a 3rd plane that not only contains the axes of the cones, but the centers of the spheres as well. The diagram shown here is what the 3rd plane looks like after intersecting the spheres and the centers as well as the cones and their axes. Quickie proof.