26*. What kind of mapping is the composition of two homotheties in space? If this mapping is a homothety, what will be its coefficient? How to determine the position of its center?
26. A homothety is fully determined by the assignment of two parallel segments $[A B]$ and $\left[A^{\prime} B^{\prime}\right]$, if it is specified that under the homothety $A$ maps to $A^{\prime}$ and $B$ to $B^{\prime}$. The center $S$ of the homothety is the intersection $S=\left(A A^{\prime}\right) \cap\left(B B^{\prime}\right)$, and the coefficient $k$ of the transformation is the ratio $\frac{\overrightarrow{A^{\prime} B^{\prime}}}{\overrightarrow{A B}}=k$.

When the points $A, B, A^{\prime}, B^{\prime}$ lie on the same line, knowing the coefficient $k$, we can draw two parallel rays from points $A$ and $A^{\prime}$, directed in the same or opposite directions depending on the sign of the coefficient $k$, and mark points $C$ and $C^{\prime}$ on these rays that satisfy the condition:

$$
\frac{\overrightarrow{A^{\prime} C^{\prime}}}{\overrightarrow{A C}}=k
$$

The position of the center in this case is determined by the intersection $S=\left(A A^{\prime}\right) \cap\left(C C^{\prime}\right)$.

Now let us consider the composition of two homotheties.

Suppose the first homothety maps $\left[A_{1} B_{1}\right]$ to $\left[A_{2} B_{2}\right]$, and the second maps $\left[A_{2} B_{2}\right]$ to $\left[A_{3} B_{3}\right]$. Since $\left(A_{1} B_{1}\right) \|\left(A_{2} B_{2}\right)$ and $\left(A_{2} B_{2}\right) \|\left(A_{3} B_{3}\right)$, we obtain that $\left(A_{1} B_{1}\right) \|\left(A_{3} B_{3}\right)$, and thus there exists a homothety that transforms the first segment into the third. It can be shown that this homothety is the composition of the first two. The exception is the case when $\overrightarrow{A_{1} B_{1}}=\overrightarrow{A_{3} B_{3}}$. In this case, the composition of the homotheties is a parallel translation

$$
\vec{m}=\overrightarrow{A_{1} A_{3}}=\overrightarrow{B_{1} B_{3}}
$$

(For this reason, sometimes a translation is considered a special case of homothety - "non-central".) The centers of these homotheties are determined as the intersections $S_{1}=\left(A_{1} A_{2}\right) \cap\left(B_{1} B_{2}\right)$ and $S_{2}=\left(A_{2} A_{3}\right) \cap\left(B_{2} B_{3}\right)$. If their composition is a homothety, then its center is $S_{3}=$ $=\left(A_{1} A_{3}\right) \cap\left(B_{1} B_{3}\right)$. If $\left|A_{2} B_{2}\right|=k_{1}\left|A_{1} B_{1}\right|$ and $\left|A_{3} B_{3}\right|=k_{2}\left|A_{2} B_{2}\right|$, then $\left|A_{3} B_{3}\right|=k_{2} k_{1}\left|A_{1} B_{1}\right|$, i.e., the coefficient of the composition is $k_{3}=k_{2} \cdot k_{1}$. The centers $S_{1}, S_{2}, S_{3}$ lie on the same line, as the triangles $A_{1} A_{2} A_{3}$ and $B_{1} B_{2} B_{3}$ satisfy the conditions of the spatial or planar Desargues theorem (see problems 4 and 5 from § 4).