Modern geometry . = -1, ACy AD_ 1 cb/ db~ AC CB AD DB PC QC OD OD PC = CQ. 58. PC = CQ Now fig. 32. PC OC CQ PC OC CQ And from similar triangles as before, AC _ PCAD~OD and {AB, CD} = -1< ^ QCDB~ CD AD~ CB CBDB AC /AD CB/ {AB, CD} is a harmonic range. Definition. If a pencil of four lines divides one transversal(and therefore every transversal*) harmonically, the pencil iscalled a harmonic pencil. 0{AB, CD} = — 1 denotes that the pencil OA, OB, OC, OD isharmonic; OC and OD are called harmonic conjugates withrespect to OA and OB. * This follows from the proposition just proved. HARMONIC
Modern geometry . = -1, ACy AD_ 1 cb/ db~ AC CB AD DB PC QC OD OD PC = CQ. 58. PC = CQ Now fig. 32. PC OC CQ PC OC CQ And from similar triangles as before, AC _ PCAD~OD and {AB, CD} = -1< ^ QCDB~ CD AD~ CB CBDB AC /AD CB/ {AB, CD} is a harmonic range. Definition. If a pencil of four lines divides one transversal(and therefore every transversal*) harmonically, the pencil iscalled a harmonic pencil. 0{AB, CD} = — 1 denotes that the pencil OA, OB, OC, OD isharmonic; OC and OD are called harmonic conjugates withrespect to OA and OB. * This follows from the proposition just proved. HARMONIC SECTION 59 Note on Theorem 27. From Theorem 26 it is easy to show that if C and D are harmonicconjugates with respect to AB, and if D is at infinity, then C is the mid-point of AB. [See also p. 8, § 3 (iii).] In the course of proving Theorem 27 we saw that a transversal PQparallel to OD is bisected by OC; it should he noticed that this is aparticular case of the theorem; for, since PQ is parallel to OD it cutsOD at infinity; therefore C and the point at infinity are har
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