Some interesting properties of Circles of Apollonius

Inspired by the problem written by David Vreken: Circles of Apollonius of a Triangle, I discovered some interesting properties of Circles of Apollonius:

If $\dfrac{|PA|}{|PB|}=\lambda\ (\lambda>0, \lambda \neq 1)$ , $O$ is the center of the Circle of Apollonius, $r$ is its radius, then we have:

1. $\dfrac{|OA|}{r}=\dfrac{r}{|OB|}=\lambda$ .

2. $|OA|\cdot |OB|=r^2$ i.e. $A$ and $B$ are a pair of inversion points of the circle.

3. $\dfrac{|OA|}{|OB|}=\lambda^2$ .

4.The intersection points of the circle and line $AB$ are the on the angle bisectors of $\angle APB$ and its supplementary angle respectively.

By the way, using these properties, solve this problem: Hardcore Analytic Geometry.

#Geometry

Easy Math Editor

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Comments

Great work! Ironically, your question In Search of a Shortcut led me down the path of investigating Circles of Apollonius in the first place.

David Vreken - 1 year, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$