PROPOSITION 12. PROBLEM.
To describe an equilateral and equiangular pentagon about a given circle.
Let ABCDE be the given circle: it is required to describe an equilateral and equiangular pentagon about the circle ABCDE.
Let the angles of a pentagon, inscribed in the circle, by the last proposition, be at the points A, B, C, D, E, so that the arcs AB, BC, CD, DE, EA are equal; and through the points A, B, C, D, E, draw GH, HK, KL, LM, MG touching the circle. [III 17.
The figure GHKLM shall be the pentagon required.
Take the centre F, and join FB, FK, FC, FL, FD.
Then, because the straight line KL touches the circle ABCDE at the point C to which FC is drawn from the centre, therefore FC is perpendicular to KL [III. 18.
therefore each of the angles at C is a right angle.
For the same reason the angles at the points B, D are right angles.
And because the angle FCK is, a right angle, the square on FK equal to the squares on FC, CK. [I. 47.
For the same reason the square on FK is equal to the squares on FB, BK.
Therefore the squares on FC, CK are equal to the squares on FB,BK; [Axiom 1.
of which the square on FC is equal to the square on FB;
therefore the remaining square on CK is equal to the remaining square on BK, [Axiom 3.
and therefore the straight line CK is equal to the straight line BK.
