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.

142

A SEQUTEL TO EUCLID.

A, E, D is equal to the anharmonic ratio of the pencil
(C. ABA'B'). Again, the points B, A, E, D are the intersections of the line AB with the pencil (C. ABA'B');
therefore the pencil (C. ABA'B') = (C'. ABA'B').
We have proved the second part of our Proposition,
and the first follows from it by the theorem of this
Article.
(2). If two triangles be such that the sides of one are the
polars of the vertices of the other, they are in perspective.
Dem. —Let the three sides of the L ABC be the
polars of the corresponding vertices of the L A'B'C',
and let the corresponding sides meet in the points X,
Y, Z respectively. Now, since AB is the polar of C',
and B'C' the polar of A, the point D is the pole of AC'
(Cor., Prop. 25, Section I., Book III.). In like manner the point X is the pole of AA', and the points B',
C' are, by hypothesis, the poles of the lines AC, AB.
Hence the anharmonic ratio of the points B', C', D, X
= the pencil (A. YZCA') = the pencil (Z. YAC'A').
Again, the anharmonic ratio B'C'DX = the pencil
(Z. A'C'AX) = (Z. XAC'A'). Hence (Z. YAC'A')
= (Z. XAC'A');.. the lines XZ, ZY form one right
A
line; therefore the intersection of corresponding sides
of the triangles are collinear. Hence they are in perspective.