---
tag: F9410EB0N1
kind: problem
name: Problem of Apollonius
msc: [51M15, 51M04]
uses: [060W5CBZP6, 0DAECMZVRT]
about: [060W5CBZP6]
sources:
  - url: https://en.wikipedia.org/wiki/Problem_of_Apollonius
same_as:
  wikidata: Q619449
  wikipedia: Problem of Apollonius
history:
  posed_by: "Apollonius of Perga"
---

## Problem

Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$ be three [[circles|060W5CBZP6]] in the Euclidean plane.
Construct, with [[straightedge and compass|0DAECMZVRT]], 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.

## Explanations

### L1 human

You are given three circles drawn on a sheet of paper.
The task is to draw a fourth circle that just touches each of the three.
There are usually several answers, up to eight, depending on which of the given circles the new circle wraps around.