Read e-book online A Sequel to the First Six Books of the Elements of Euclid, PDF

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EF . PG = AP . PB . but This Prop. and its Demonstration will hold true when the 0 s are external to each other. Oor. If AB be the diameter of the 0 ABO, this Prop. reduces to the last. 0 . ' 0 0 Prop . -To draw a oommon tangent to two Let P be the centre of the greater 0 , Q the centre of the less, with P as centre, and a radius = to the dif­ ference of the radii of the two 0 8 : de­ scribe the 0 IGR ; from Q dr aw a tangent E olroles. D to thi s 0 , touching it at H. A SEQUEL TO EUCLID . Join PH, and produce it to meet the circumference of the larger 0 in E .

C - BC) = 2AB ED. -If A, B, C, D befourpoints taken in order on a right It'n e, then AB CD A B e D + B C . AD = AC . BD. · I I I I Dem,--Let AB a, B C = b, CD = 0 ; then All . CD + BC AD = ao + b (a + b + o) = (a + b) (b + c) = AC . B D . = This theorem, which is due to Euler, is one of the most important in Elementary G eometry. It may be w ri tt en in a more symmetrical form by making use 24 A SEQUEL TO EUCLID . of the convention regarding plus and minus : thus, since + AC = - CA, we get AB .

Prop. -If two circles touch each other at any point P, and any line out the o�'r oles �·n P the points A, B, 0, D ; then the :ID (tngle APB = OPD. ) L E PA = PDA. Hence, by subtraction, L AP13 OPD Prop. -If a oirole touoh a semioirole �'n D ana its d�'ameter in P, and PE be per­ pendioular to the diameter at P, the square on PE is equal to tw�'oe the reotang le oontaz'n ed by the radii of the oiroles. -Oomplete the circle, A and produce EP to meet it again in G. Let 0 and F be the centres ; then the line OF will pass through D .

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