Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/138

114 6. A circle is said to be described about a rectilineal figure, when the circircumference of the circle passes through all the angular points of the figure about which it is described.

7. A straight line is said to be placed in a circle, when the extremities of it are in the circumference of the circle.

PROPOSITION 1. PROBLEM.

In a given circle, to place 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: it is required to place in the circle ABC, a straight line equal to D.

Draw BC, a diameter of the circle ABC. Then, if BC is equal to D, the thing required is done; for in the circle ABC, a straight line is placed equal to D. But, if it is not, BC is greater than D. [Hypothesis. Make CE equal to D, and from the centre C, at the distance CE, describe the circle AEF, and join CA.

Then, because C is the centre of the circle AEF, CA is equal to CE; [I. Definition 15. but CE is equal to D; [Construction. therefore CA is equal to D. [Axiom 1.

Wherefore, in the circle ABC, a straight line CA is placed equal to the given straight line D, which is not greater than the diameter of the circle.