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Here is the background to my question:

A mathematician thinking about Euclidean geometry finds she is studying those properties of figures which are left unchanged by the elements of a particular group, the group of similarities for the plane. A similarity enlarges or shrinks figures while keeping them in the same shape. More precisely, it sends straight-line segments to straight-line segments, multiplying the length by a factor which is the same for every segment. Triangles are sent to similar triangles, angles being preserved in magnitude, but not necessarily in the sense. The composition of two similarities is another. This forms a group.

This seems obvious to me, yet in order to prove this I am unsure what sort of function is needed. How can I send straight-line segments to straight-line segments and preserve angles in a triangle?

A hint is more appreciated than a solution.

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    There are groups of rotations, translations, and scalings. Combine them to make a bigger group. – 2017-01-03

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Hint : Suppose you have two lines in the plane $L_1$ joining $(0,0)$ and $(4,3)$ and $L_2$ joining $(0,0)$ and $(2,2)$.(Draw figures !!) How will you map one onto the other ?

The length of $L_1$ is $5$ and that of $L_2$ is $2\sqrt2$, so first of all you'll multiply the coordinates for $L_2$ by $(\frac{5}{2\sqrt2})^2$. Then you will possibly rotate or translate or reflect the line till it finally superimposes on $L_1$. This operation of scaling and then applying either translation, reflection etc. forms the "group of similarities" of the plane, i.e., composition of scaling with the isometries of the plane.

Similarly similar triangles can be mapped to each other by appropriate scaling and then applying isometries of the plane.

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    But is there a way to do all of that "translating, scaling, rotating" at once? In other words, is there a function that can do all this? – 2017-01-11
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    @JacopoStifani I don't think so. – 2017-01-11