### Proposition 1.

*
Into a given circle to fit a straight line equal to a given straight line
which is not greater than the diameter of the circle.
*

Let ABC be the given circle, and D the given straight line not greater than
the diameter of the circle; thus it is required to fit into the circle ABC a
straight line equal to the straight line D.

Let a diameter BC of the circle ABC be drawn.

Then, if BC is equal to D, that which was enjoined will have been done; for BC
has been fitted into the circle ABC equal to the straight line D.

But, if BC is greater than D, let CE be made equal to D,

and with centre C and distance CE let the circle EAF be described;

let CA be joined.

Then, since the point C is the centre of the circle EAF, CA is equal to CE.
But CE is equal to D; therefore D is also equal to CA.

Therefore into the given circle ABC there has been fitted CA equal to the
given straight line D.

Q.E.F.

Ralph H. Abraham, 28 April, 1996.