Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/355

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point; let any chord of the circle be drawn so that, produced if necessary, it may pass through B. Let P be the middle point of this chord, so that P is a point on the required locus.

The straight line AP is at right angles to the chord of which P is the middle point (III. 3); therefore P is on the circumference of a circle of which AB is a diameter. Hence if B be within the given circle the locus is the circumference of the circle described on AB as diameter; if B be without the given circle the locus is that part of the circumference of the circle described on AB as diameter, which is within the given circle.

52. O is a fixed point from which any straight line is drawn meeting a fixed straight line 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 shew that the locus of Q is a straight line.

For draw a perpendicular from O on the fixed straight line, meeting it at C; in 0C take a point D such that OD is to OC in the fixed ratio; draw from O any straight line OP meeting the fixed straight line at P, and in OP take a point Q such that OQ is to OP in the fixed ratio; join

QD. The triangles ODQ and OCP are similar (VI. 6); therefore the angle ODQ is equal to the angle OCP, and is therefore a right angle. Hence Q lies in the straight line drawn through D at right angles to OD.