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

 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. From the centre A at the distance AB describe the circle BCD. From the centre B, at the distance BA, describe the circle ACE. From the point C, at which the circles cut one another, draw the straight lines CA and CB to the points A and B. ABC shall be an equilateral triangle. Because the point A is the centre of the circle BCD, AC is equal to AB. And because the point B is the centre of the circle ACE, BC is equal to BA. 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. Therefore CA is equal to CB. Therefore CA, AB, BC are equal to one another. Wherefore the triangle ABC is equilateral, and it is described on the given straight line AB. Q.E.F.