Problem of Apollonius

Builds on Circle Straightedge and compass construction

Problem

Let $\Gamma_1$ , $\Gamma_2$ , $\Gamma_3$ be three circles in the Euclidean plane. Construct, with straightedge and compass, every circle that is tangent to all three of them.

Remarks

The problem is solved: straightedge and compass constructions of all solutions are known. Apollonius posed it in his work Tangencies, which is lost, so his own solution is not known in detail. François Viète gave a straightedge and compass solution. A solution circle may touch each $\Gamma_i$ from outside or from inside, which gives $2^3 = 8$ cases. So, outside special positions with infinitely many solutions, there are at most eight solution circles, and some positions give fewer or none. Inversion in a circle and the algebra of three quadratic equations both lead to solutions. Variants replace some of the circles by points or by lines.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Circle concept
Straightedge and compass construction concept
Problem of Apollonius problem

Tag	`F9410EB0N1`
URI	`m2://F9410EB0N1`
Kind	problem
Writing	W0 stub
MSC	51M15, 51M04
Status	This copy of the site has no build data. Run `m2 status F9410EB0N1`.
About	Circle
Uses	Circle, Straightedge and compass construction
Same as	Wikidata Q619449 Wikipedia Problem of Apollonius

Problem of Apollonius

Problem

Remarks

Reading path

Further reading

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