Connect the point $P$ on the circle with the fixed point $A$ inside the circle. Divide the distance $PA$ in the ratio $m: n$. What will be the geometric locus of the division point as $P$ moves along the circle?
The distance $P A$ is divided into two parts by the point $M$ such that

$$
P M: M A=m: n
$$

![](https://cdn.mathpix.com/cropped/2024_05_02_cd5aa022a6e5676c174bg-1.jpg?height=474&width=469&top_left_y=327&top_left_x=817)

Draw a line parallel to $P O$ from the point $M$; this line intersects $A O$ at the point $O'$. Then

$$
\begin{aligned}
& M O': P O = M A: P A = n: (m+n) \\
& M O' = \frac{n}{m+n} \quad P O = \frac{n}{m+n} r \ldots
\end{aligned}
$$

if, for instance, $P O = r$ is the radius of the given circle.

$O'$ is a fixed point on the distance $A O$, because

$$
A O': A O = A M: A P = n: (m+n) \ldots
$$

According to 1), the distance of the point $M$ from the fixed point $O'$ is constant: the geometric locus of the point $M$ is a circle whose center is $O'$, which divides the distance $A O$ in a fixed ratio, and whose radius is: $\frac{n}{m+n} r$. This circle is in a similar position to the original circle with respect to the given point $A$. (External similarity point!)

Alfréd Weisz (Bolyai g. VIII. o. Bp. V.)

Note. We arrive at the same result if $P$ is outside the circle. (Weisz A.)