Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/31

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PROPOSITION 1. PROBLEM.

To describe an equilateral triangle on a given finite straight line.

Let AB be the given straight line; it is required to describe an equilateral triangle on AB.

Equilateral Triangle Construction.svg

From the centre A at the distance AB describe the circle BCD. [Postulate 3.

From the centre B, at the distance BA, describe the circle ACE. [Postulate 3.

From the point C, at which the circles cut one another, draw the straight lines CA and CB to the points A and B. [Post. 1.

ABC shall be an equilateral triangle.

Because the point A is the centre of the circle BCD, AC is equal to AB. [Definition 15.

And because the point B is the centre of the circle ACE, BC is equal to BA. [Definition 15.

But it has been shewn that CA is equal to AB;

therefore CA and CB are each of them equal to AB.

But things which are equal to the same thing are equal to one another. [Axiom 1.

Therefore CA is equal to CB.

Therefore CA, AB, BC are equal to one another.

Wherefore the triangle ABC is equilateral, [Def. 24. and it is described on the given straight line AB. q.e.f.