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In the configuration below, four circles $C_i$, $i=1,2,3,4$, are tangent as shown, and each tangent to a surrounding circle $C_0$.

Q. Are the four circle intersections shown cocircular?

     4Circles

Joseph O'Rourke
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  • What makes you think this is true? – Parcly Taxel May 12 '21 at 11:56
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    This is a known result. By using the fact that the opposite angles in a cyclic quadrilateral are supplementary and some angle chasing it can be proven. Alternatively, by inversion (see this link: https://www.cut-the-knot.org/Curriculum/Geometry/FourTouchingCircles.shtml) it can also be proven. The tangency to the surrounding circle condition is unnecessary. In the link I provided, there is also a generalization for cases where the number of external tangencies are even. – krazy-8 May 12 '21 at 12:10
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    @dodoturkoz: Great, thanks! – Joseph O'Rourke May 12 '21 at 12:16

1 Answers1

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Inversion of this picture centered at the touch point between circles $1$ and $3$ transforms this picture into two parallel lines with three equal size circles in between them, touching both lines and each other (in the order $2$, $0$, $4$). Clearly in that picture these four points are on a circle (they are the points where the two outer circles touch the parallel lines, so they form a rectangle). Then they are in the original picture as well.

It is not relevant that circles $2$ and $4$ also touch some circle $0$. It is sufficient that they touch both $1$ an $3$

WimC
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