## Task 2 - 070822

Given are a circle $k$ (center $M$, radius of length $r=6 \mathrm{~cm}$) and a circle $k_{1}$ (center $M_{1}$, radius of length $r_{1}=2 \mathrm{~cm}$). Both circles touch each other externally.

Construct all circles with a radius of length $2 \mathrm{~cm}$ that touch the two given circles!

Also construct the points of contact of the sought circles with the given ones!
(I) The centers of the sought circles lie either (1) on the circle around $M$ with the radius of length $r+r_{1}$ or (1') on the circle around $M$ with the radius of length $r-r_{1}$, since they are supposed to touch $k$, and (2) on the circle around $M_{1}$ with the radius of length $r+r_{1}$, since they are supposed to touch $k_{1}$ and have the same radius as $k_{1}$.

(II) Therefore, the centers of the sought circles result from the intersection points of the circles according to (1) and (2) (these are 2 intersection points) and as the point of tangency of the circles according to (1') and (2).

This point lies on the line connecting $M$ and $M_{1}$. The points of tangency of the circles are obtained by connecting the centers of the touching circles. There are exactly 3 circles that satisfy the conditions of the problem. The proof follows from (I).