English: Illustration of a proof of Apollonius' definition of a circle.

Apollonius discovered that a circle could be defined as the set of points ${\displaystyle P}$ that have a given ratio of distances to two given points ${\displaystyle A}$ and ${\displaystyle B}$, so ${\displaystyle {\frac {|AP|}{|BP|}}={\text{constant}}}$.

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 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 ${\displaystyle AB}$, which satisfies the ratio. So $${\displaystyle {\frac {|AP|}{|BP|}}={\frac {|AD|}{|BD|}}.}$$

Also take some other point ${\displaystyle Q}$ anywhere on the extended line ${\displaystyle AP}$. Also by the Angle bisector theorem the line ${\displaystyle AD}$ bisects the exterior angle ${\displaystyle \angle QPB}$. Hence, ${\displaystyle \gamma =\angle QPD}$ 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.