Constructing an equilateral triangle

In this exercise you will construct an equilateral triangle using a ruler and compass. Given two points \(X\) and \(Y\), can you construct a third point \(Z\) such that \(|XY| = |XZ| = |YZ|\)?

As an example, the diagram below contains two points \(A\) and \(B\), from which a third point \(C\) has been constructed such that \(|AB| = |AC| = |BC|\). Your task is to figure out how to do this construction. You can click and drag the points \(A\) and \(B\), and the point \(C\) will automatically be redrawn so that the equality \(|AB| = |AC| = |BC|\) remains true.

In the applet below you can draw lines, segments and circles, as well as creating a new point at the intersection of two lines, a line and a circle or two circles. At the moment the tools you can use are limited, since all you have is a ruler and compass. To create more tools you will need to complete more of the exercises!

Hint. This exercise is explained in Euclid Book I, Proposition 1.