158 
Dr Taylor, Geometrical Notes on Theorems, etc. 
h. Or in the circle, supposing that I is not the centre, draw 
a fixed chord HIK, and let the parallel through I to OK meet 
OP, OH, OQ in x, n, z. 
Then by the circles IHPx, IHzQ, 
Z xHP = Z xIP = z.QIz = Z. QHz. 
Hence nx.nz — nO.nH, 
the points 0, x, H, z lying on a circle because Z xHz = 7 r — 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 = a constant. 
This constant may be determined by making OP, OQ coincide 
with 01 and the tangent at 0. 
When 01 is the normal at 0 the constant is On 2 , and OP, OQ 
are always at right angles. 
Hence Fregier’s theorem and its converse for the ellipse. 
