C.a.R. > Applications > Difficult > Touching three Circles

How to find the red circle? The problem is due to Apollonius of Perge as many other problems and results too. It was treated and solved by Newton. The solution given here uses reflection at a circle.

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• First the black circles are enlarged, such that two of them touch. You get the green circles.
• Then the all green circles are reflected at the blue circle. We get two brown lines and a brown circle.
• By elementary constructions, we construct a red circle, touching all brown objects.
• Reflecting that circle once more and enlarging it yields the solution.

Here is a similar construction.

Technical Information

Most steps are completely hidden.