Modern geometry . is harmonic. [Through C and C draw parallels to OD, and use Exs. 222, 223.] Definition. A system of lines through a point is called apencil. The point is called the vertex of the pencil. Definition. Any straight line drawn across a system of linesis called a transversal. HARMONIC SECTION 57 Theorem 27. If a transversal cuts the four lines of a pencil atA, c, B, D, and if AC BD is a harmonic range, then anyother transversal will also be divided fig. 32. Let O be the vertex of the pencil. Let a straight line cut the rays of the pencil at A, C, B, D. Through C and
Modern geometry . is harmonic. [Through C and C draw parallels to OD, and use Exs. 222, 223.] Definition. A system of lines through a point is called apencil. The point is called the vertex of the pencil. Definition. Any straight line drawn across a system of linesis called a transversal. HARMONIC SECTION 57 Theorem 27. If a transversal cuts the four lines of a pencil atA, c, B, D, and if AC BD is a harmonic range, then anyother transversal will also be divided fig. 32. Let O be the vertex of the pencil. Let a straight line cut the rays of the pencil at A, C, B, D. Through C and C draw straight lines || to OD, cutting OAat P, P and OB at Q, Q. •/ PC is II to OD. .. As APC, AOD are similar. AC _ PC •• AD~OD also As BQC, BOD are similar. CBQC t , 1 DB~OD* •.•{AB, CD} = -1 „? , , , .PC = CQ ^ Since {AB, CD} = -1, ACy AD_ 1 cb/ db~ AC CB AD DB PC QC OD OD PC = CQ. 58
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