A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

BOOK VI.

93

contained by one pair of opposite sides is equal to the
rectangle contained by the other pair.
Prop. 10.-If through any point 0 two lines be drawn
cutting a circle in four points, then joining these points
both directly and transversely; and if the direct lines meet
in P and the transverse lines meet in Q, the line PQ will
be the polar of the point 0.
D
Dem.-Join OP; then the pencil (P. OAEB) is harmonic (Prop. 7);.. the points 0, E are harmonic
conjugates to the points A, B. Hence the polar of 0
passes through E (Prop. 4). In like manner, the
polar of 0 passes through F;.*. the line PQ, which
passes through the points E and F, is the polar of 0.
Q.E.D.
Cor. 1.-If we join the points 0 and Q, it may be
proved in like manner that OQ is the polar of P.
Cor. 2.-Since PQ is the polar of 0, and OQ the
polar of P, then (Cor. 1, Prop. 16, Section I., Book III.)
OP is the polar of Q.
DEF.-Triangles such as OPQ, which possess the property that each side is the polar of the opposite angular
point with respect to a given circle, are called self-conjzgate triangles with respect to the circle. Again, if we