53. O is a fixed point from which any straight line is drawn meeting the circumference of a fixed circle at P ; in OP a point Q is taken such that OQ is to OP in a fixed ratio: determine the locus of Q.
We shall show that the locus is the circumfercuce of a circle.
For let C be the centre of the fixed circle; in OC take a point D such that OD is to OC in the fixed ratio, and draw any radius CP of the fixed circle; draw DQ parallel to CP meeting OP, produced if necessary, at Q. Then the triangles OCP and ODQ are similar (VI. 4), and therefore OQ is to OP as OD is to 0C, that is, in the fixed ratio. Therefore Q is a point on the locus. And DQ is to CP in the fixed ratio, so that DQ is of constant length. Hence the locus is the circumference of a circle of which D is the centre.
54. There are four given points A, B, C, D in a straight line; required the locus of a point at which AB and CD subtend equal angles.
Find a point O in the straight line, such that the rectangle OA, OD may be equal to the rectangle OB, OC (34), and take OK such that the square on OK may be equal to either of these rectangles (II. 14): the circumference of the circle described from as centre, with radius OK shall be the required locus.
[We will take the case in which the points are in the following order, O, A, B, C, D.]
For let P be any point on tho circumference of this
