APPENDIX |
331 |

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*.