A Sequel to the First Six Books of the Elements of Euclid, by John Casey

By John Casey

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Extra info for A Sequel to the First Six Books of the Elements of Euclid, Fifth Edition

Example text

And 11. iii. EP 2 = DF . DH = twice rectangle contained by the radii. Prop. -If a m;rcle PGD touoh a oirole AB C in D and a chord AB in P, and tt/ EF be perpendioular to AB , at its middle point, and at the 8tt'ae opposite to that of D the o£rcle PGD, the rectangle contatt'ned by EF and the dtt'ameter ofthe circle PGDis equal At--��--+E-----J- B to the rectangle AP . PB. -Let PG be at right L s to AB, then PG is the diameter of the 0 H F PGD . Join DG, DP, and produce them to meet the 0 AB C in 0 and F ; then OF is the diameter of the 0 ABC, and is 11 to PG (2) ; OF is J..

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 . Draw QF 11 to PE. Join EF, which will be the common tangent required. -The lines HE and QF are, from the construc­ tion, e qual ; and since they are 11, the fig. HEF Q is a c:J ; . ' . the L PEF = PHQ = right angle ; . EF is a tangent at E ; and since L EFQ = EHQ � right angle, EF is a tangent at F. The tangent EF is called a d�'reot com­ mon tangent. If with P as centre, and a radius equal to the sum of the radii of the two given 0 s, we shall describe a <:) , we shall have a common tangent which will pass between the 0 s, and one which is called a transverse common tangent.

PFD + PFE = PAO + PAE = two right L s. Hence the points D, F, E, are collinear. Cor. -If the feet of the l... s drawn from any point P to the sides of the � ABO be B ¥----ll-----y collinear, the locus of P is the o described about the triangle. Cor. -If four lines be given, a point can be found su ch, that the feet of the B four l... s from it on the lines will be collinear. For let the four lines be AB, AC, DB, DF. These lines form four �s. Let the 0 s described about two of the � s-say AFE, ODE-intersect in P ; then it is evident that the feet of the 1...

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