<s id="A18-1.14.01">[14] After having said this in advance, we are going to prove that we can find, for any given figure, a similar one that is in a given proportionality to it.</s>
<s id="A18-1.14.02">We shall prove this first for the plane.</s>
<s id="A18-1.14.03">Let us assume any line, namely the line <ab>, that is fixed at point <a> and moves in a plane.Let there be two points on it, namely the points <b>, <h>, that move with the line.</s>
<s id="A18-1.14.04">Point <b> describes in the plane the (circle-)line <bgdez> and point <h> the (circle-)line <hqklm>, so we say that the two (circle-)figures <bgdez> and <hqklm> are similar to one another.</s>
<s id="A18-1.14.05">Proof: Let us draw into <bgdez> a figure of straight lines, namely the figure <bgdez>; let us further draw the figure <hqklm> by drawing lines from point <a> to the points <bgdez>, namely the lines that we have already drawn; let us further connect the points <hqklm>, then, because the lines <ba>, <ga>, <da>, <ea>, <za> are, according to our assumption, similarly divided at the points <hqklm>, the one figure is of straight lines, namely <bgdez>, similar to the other figure of straight lines, namely <hqklm>.</s>
<s id="A18-1.14.06">In a similar way we prove that we can draw inside the figure <hqklm> a figure of straight lines that is similar to any (arbitrary) figure of straight lines drawn inside <bgdez>, because the figures described by the two points are similar.</s>
