Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/354

330 if necessary, at D. Then the square on AC is equal to the squares on AD and CD, and the square on BC is equal to the squares on BD and CD (I. 47); therefore the. square on AC exceeds the square on BC by as much as the square on AD exceeds the square on BD. Hence D is a fixed point either in AB or in AB produced through B (40). And the required locus is the straight line drawn through D, at right angles to AB.

50. Required the locus of a point such that the straight lines drawn from it to touch two given circles may be equal

Let A be the centre of the greater circle, B the centre of a smaller circle; and let P denote any point on the required locus. Since the straight lines drawn from P to touch the given circles are equal, the squares on these straight lines are equal. But the squares on PA and PB exceed these equal squares by the squares on the radii of the respective circles. Hence the square on PA exceeds the square on PB, by a known square, namely a square equal to the excess of the square on the radius of the circle of which A is the centre over the square on the radius of the circle of which B is the centre. Hence, the required locus is a certain straight line which is at right angles to AB (49).

This straight line is called the radical axis of the two circles.

If the given circles intersect, it follows from III. 36, that the straight line which is the locus coincides with the produced parts of the common chord of the two circles.

51. Required the locus of the middle points of all the chords of a circle which pass through a fixed point.

Let A be the centre of the given circle; B the fixed