Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/146

122 PROPOSITION 9. PROBLEM.

To describe a circle about a given square.

Let ABCD be the given square: it is required to describe a circle about ABCD.

Join AC, BD, cutting one another at E.

Then, because AB is equal to AD, and AC is common to the two triangles BAC, DAC; the two sides BA, AC are equal to the two sides DA,AC each to each; and the base BC is equal to the base DC; therefore the angle BAC is equal to the angle DAC, [I. 8. and the angle BAD is bisected by the straight line AC.

In the same manner it may be shewn that the angles ABC, BCD, CDA are severally bisected by the straight lines BD, AC.

Then, because the angle DAB is equal to the angle ABC, and that the angle EAB is half the angle DAB, and the angle EBA is half the angle ABC, therefore the angle EAB is equal to the angle EBA; [Ax. 7. and therefore the side EA is equal to the side EB. [I. 6.

In the same manner it may be shewn that the straight lines EC, ED are each of them equal to EA or EB. Wherefore the four straight lines EA, EB, EC, ED are equal to one another, and the circle described from the centre E, at the distance of any one of them, will pass through the extremities of the other three, and will be described about the square ABCD.

Wherefore a circle has been described about the given sqaure.

PROPOSITION 10. PROBLEM.

To describe an isosceles triangle, having each of the angles at the base double of the third angle.