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Exercises on Euclid and in modern geometry for the use of schools, private students, and junior university students (1881)

McDowell, James L · Cambridge, Deighton, Bell, and Co. · leaf 243

An illustration from Exercises on Euclid and in modern geometry for the use of schools, private students, and junior university students, 1881, leaf 243 — compounding these ratios, we have, EP2 : PM2 :: CE.DE : KM. LM :: CE. CF : KM. KX :: CA2 : AK2. Therefore CA : AK :: ^P : PJf. But C^l : AK :: .#£ : £J/, and therefore EG : £Jf :: EP : PJf. Therefore EM is cut harmonically, for it is cut internally andexternally in the same ratio. Q. e. d. 214… … Fig. i. Let C be the middle point of the chord AB, and through Cdraw any other two chords BE and FG; then shall AH and BKbe equal. First, by pencils, Fig. 1. {D.AGFB} = {F.AGFB}, (189). Therefore [AHCB] = [ACKB], and therefore

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