Advanced geometry for high schools : synthetic and analytical (1919)
McDougall, A. H. (Alexander Hiram), 1859-1946 · Toronto : Copp, Clark · leaf 58
= BF. HARMONIC RANGES AND PENCILS 45 (43) By a proof similar to that of § 42, the followingconverse to the theorem of that article may be shownto be true. If, in the pencil p(a, c, b, D), a straight linethrough B parallel to PA cut PC, PD at E, F re-spectively such that be = bf, then P(a, c, b, d)… … Fia. 45. It is required to show that K, M, N, L is a harmonicrange. Through B, N respectively draw EF, GH || PA. V P(A, C, B, D) is a harmonic pencil, and EBF || AP .-., by § 42, EB = BF. ™ • 1 a GN EB 1 NP BP From similar As, -^ = — and — = —, .-., by multiplication, GN : NH = EB : BF, .-. GN =…
Geometry · Geometry, Analytic
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