Apollonius circle

Apollonius circle — a similarity tool that converts "weighted distance" into "unit-weight distance".

Universal form

$\min\big(\, PA + k \cdot PB \,\big), \quad 0 < k < 1$

Classical model

• Moving point $P$: on a circle $\odot C$ (radius $r$)
• Fixed point $A$: anywhere outside the circle
• Fixed point $B$: outside the circle, distinct from the centre $C$
• Key constraint: $k = \dfrac{r}{CB}$ (if not, rescale the figure first via similarity; otherwise this is not an Apollonius-circle problem)

Construction and proof

1. On ray $CB$, take an auxiliary point $B'$ with

   $CB' = \dfrac{r^2}{CB} \quad\Leftrightarrow\quad CB' \cdot CB = r^2 = CP^2$

   

2. By $\dfrac{CB'}{CP} = \dfrac{CP}{CB} = k$ and the shared angle $\angle BCP$,

   $\triangle CPB' \sim \triangle CBP \quad (\text{SAS})$

   

3. The similarity ratio gives $\dfrac{PB'}{PB} = k$, so

   $PB' = k \cdot PB$

   for every $P$ on the circle — this is the key invariant.

4. Therefore $PA + k \cdot PB = PA + PB' \geq |AB'|$ ([[triangle-inequality]]), with equality when $A, P, B'$ are collinear and $P$ lies between $A$ and $B'$.

   

Answer

$\min\big(\, PA + k \cdot PB \,\big) = |AB'|$

where $|AB'|$ is computed directly from the geometry of $\triangle ACB'$.

Applications

• [[0003-apollonius-circle]] — the standard right-triangle-plus-circle problem