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To inscribe a Square in a Circle.
TO inscribe a Square in the Circle ABCD; draw to the Diameter AB, the Perpendicular DC, which may pass through the Center E. Draw also the Lines AC, CB, BD, AD; and you will have inscribed in the Circle the Square ACBD.
Demonstration. The Triangles AEC, CEB, have their Sides equal, and the Angles AEC, CEB, equal, seeing they are Right, therefore the Bases AC, CB, are equal (by the 4th. of the 1st.) Moreover, seeing the Sides AE, CE, are equal, and the Angle E being Right, they shall each of them be semi-right, (by the 32d. of the 1st.) So then the Angle ECB is semi-right. And by consequence the Angle ACB shall be Right. It is the same of all the other Angles: there∣fore the Figure ACDB is a Square.