Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/352

328 46. In a given circle it is required to inscribe a tri- angle so that the sides may pass through three given points.

Let A, B, C be the three given points. Suppose PMN to be the required triangle inscribed in the given circle

Draw NE parallel to AB, and determine the point F as in the preceding problem. We shall then have to describe in the given circle a triangle EMN so that two of its sides may pass through given points, F and C, and the third side be parallel to a given straight line AB. This can be done by the preceding problem.

This example and the preceding are taken from the work of Catalan already cited. The present problem is sometimes called Castillon's and sometimes Cramer's; the history of the general researches to which it has given rise will be found in a series of papers in the Mathematician Vol. Ill, by the late T. S. Davies.

ON LOCI.

47. A locus consists of all the points which satisfy certain conditions and of those points alone. Thus, for example, the locus of the points which are at a given distance