Properties of proportional segments

Parallel lines cutting both sides of an angle make proportional segments on the sides: AB₁ : AC₁ = B₁B₂ : C₁C₂ = B₂B₃ : C₂C₃. The proof uses similar triangles, since the corresponding angles are equal, together with the equal opposite sides of parallelograms. From this we get a way to divide a segment in the ratio m : n : k: put segments of lengths m, n, k one after another on a side of an acute angle, join the last point to the end of the given segment on the other side, and draw parallels to that line. In the same way we construct the fourth proportional segment d for a : b = c : d: lay off OA = a, AB = b and OC = c, draw through B the parallel to AC, and CD = d. A compass point is sharp, so be careful when constructing.