Problem. In a given circle (ABCD) to inscribe a square.
Sol.—Draw any two diameters AC, BD at right angles to each other. Join AB, BC, CD, DA. ABCD is a square.
Dem.—Let O be the centre. Then the four angles at O, being right angles, are equal. Hence the arcs on which they stand are equal [III. xxvi.], and
hence the four chords are equal [III. xxix.]. Therefore the figure ABCD is equilateral.
Again, because AC is a diameter, the angle ABC is right [III. xxxi.]. In like manner the remaining angles are right. Hence ABCD is a square.