Constructing an equilateral triangle

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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.