Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/144

120 PROPOSITION 7. PROBLEM.

To describe a square about a given circle.

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

Draw two diameters AC, BD of the circle ABCD, at right angles to one another; [III. 1, 1. 11. and through the points A,B,C,D, draw FG, GH, HK, KF touching the circle. [III. 17. The figure GHKF shall be the square required.

Because FG touches the circle ABCD, and EA is drawn from the centre E to the point of contact A, [Construction. therefore the angles at A are right angles. [III. 18. For the same reason the angles at the points B, C, D are right angles. And because the angle AEB is a right angle, [Construction. and also the angle EBG is a right angle, therefore GH is parallel to AC. [I. 28. For the same reason AC is parallel to FK.

In the same manner it may be shewn that each of the lines GF, HK is parallel to BD. Therefore the figures GK, GC, CF, FB, BK are parallelograms; and therefore GF is equal to HK, and GH to FK. [I. 34.

And because AC is equal to BD, and that AC is, equal to each of the two GH, FK, and that BD is equal to each of the two GF, HK, therefore GH, FK are each of them equal to GF, or HK; therefore the quadrilateral figure FGHK is equilateral.

It is also rectangular. For since AEBG is a parallelogram, and AEB a right angle, therefore AGB is also a right angle. [I. 34. In the same manner it may be shewn that the angles at H, K, F are right angles;