Elementary Projective Geometry

Cover Elementary Projective Geometry
Elementary Projective Geometry
Alfred Garside Pickford
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= (f>, we must have tan (6 + a) tan (^ — a) = k, tan^ — tan^ a = ^ (1 — tan^ . Tan^ a), or tan^^ (1 +^tan"^a) = A; + tan^a; 1 — tan^ a ■•*-^^=tan^. (l-T + k tan^ a/ ' this gives two real and different, coincident, or imaginary double rays, according as 1 — tan^ a I 1 + A; tan^ a, I. E. As k^ — tan^ a.
Projective Rows and Pencils 57 EXAMPLES. III.
1. A line KL parallel to AB cuts TA, TB at K, L, and P is taken on TA. By means of a common perspective row find Q on TB, such that {TBLQ} = {ATKP}.
2
.... If K, L are the mid-points of TA, TB respectively, find a point Q on TB corresponding to any point P on AT, so that {TBLQ} may be equal to {ATKP}. Also prove that AP : PT as TQ : QB.
3. If K, K' and L, L' be corresponding pairs of points on two projective rows KL, K'L', prove that the locus of the intersection of KL' and K'L is a straight line ; and find where this line cuts the two rows.
4. If /e, Hn a pencil correspond to k\ V respectively in another pencil projective with the former one, prove that the join of the intersections of k, V and k', I passes through a fixed point.


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