An elementary treatise on modern pure geometry (1893)
Lachlan, Robert · London, New York, Macmillan and Co. · leaf 67
in the points A, A, B, &c. Then if the pencil be in involution, we have by § 89,^n AOB. sin AOB _ sin^PC. sin ^PCsin AOB. sin AOF ~ sin AOG. sin AOG 50 PROPERTIES OF A PENCIL IN INVOLUTION. Let OiVbe the perpendicular from 0 on the line AA. Thenwe have NO.AB sin A OB =sin AOB = OA.OB NOj_ABOA .… … AG. AC Hence, we obtain the relation AB.AB _ AB.AB AC.AC- Therefore, by § 76, the range {AA, BB, GG} is in involution. Conversely, it may be proved in a similar manner that if thepoints of a range in involution be joined to any point not on thesame straight line, these lines will form a system of…
Geometry
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