158 



JDr Taylor, Geometrical Notes on Theorems, etc. 



b. Or in the circle, supposing that / is not the centre, draw 

 a fixed chord HIK, and let the parallel through / to OK meet 

 OP, OH, OQ in x, n, z. 



Then by the circles IHPcc, IHzQ, 



Z xHP = z xIP = zQIz = z QHz. 



Hence nx.nz = nO. nH, 



the points 0, x, H, z lying on a circle because Z xHz = 77- — xOz. 

 Or if these points lie on a circle, PIQ will be a straight line. 



c. Supposing HIK to be a diameter of the circle, make 

 the same construction in an ellipse with OK parallel to an axis. 



Then, by orthogonal projection, nx.nz = « constant. 



This constant may be determined by making OP, 0Q coincide 

 with 01 and the tangent at 0. 



When 01 is the normal at the constant is On 2 , and OP, OQ 

 are always at right angles. 



Hence Fregier's theorem and its converse for the ellipse. 



