Circles of Apollonius

The circles of Apollonius are any of several sets of circles associated with Apollonius of Perga, a renowned Greek geometer. Most of these circles are found in planar Euclidean geometry, but analogs have been defined on other surfaces; for example, counterparts on the surface of a sphere can be defined through stereographic projection.

The main uses of this term are fivefold:

A circle is usually defined as the set of points P at a given distance r (the circle's radius) from a given point (the circle's center). However, there are other, equivalent definitions of a circle. Apollonius discovered that a circle could be defined as the set of points P that have a given ratio of distances k = ⁠d₁/d₂⁠ to two given points (labeled A and B in the figure). These two points are sometimes called the foci.

Let d₁, d₂ be non-equal positive real numbers. Let C be the internal division point of AB in the ratio d₁ : d₂ and D the external division point of AB in the same ratio, d₁ : d₂.

Then,

Therefore, the point P is on the circle which has the diameter CD.

First consider the point ${\displaystyle C}$ on the line segment between ${\displaystyle A}$ and ${\displaystyle B}$, satisfying the ratio. By the definition $${\displaystyle {\frac {|AP|}{|BP|}}={\frac {|AC|}{|BC|}}}$$ and from the converse of the angle bisector theorem, the angles ${\displaystyle \alpha =\angle APC}$ and ${\displaystyle \beta =\angle CPB}$ are equal.

Next take the other point ${\displaystyle D}$ on the extended line ${\displaystyle AB}$ that satisfies the ratio. So $${\displaystyle {\frac {|AP|}{|BP|}}={\frac {|AD|}{|BD|}}.}$$ Also take some other point ${\displaystyle Q}$ anywhere on the extended line ${\displaystyle AP}$. Again by the converse of the angle bisector theorem, the line ${\displaystyle PD}$ bisects the exterior angle ${\displaystyle \angle QPB}$. Hence, ${\displaystyle \gamma =\angle BPD}$ and ${\displaystyle \delta =\angle QPD}$ are equal and ${\displaystyle \beta +\gamma =90^{\circ }}$. Hence by Thales's theorem ${\displaystyle P}$ lies on the circle which has ${\displaystyle CD}$ as a diameter.