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Additional resources for A Sequel to the First Six Books of the Elements of Euclid, Fifth Edition

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PK = leH . Hence RH = RE, and PK = RE. Again, since OF = FG, and FE common, L GEF = OEF ; but GEF = KEH = KHE ; . '. L OEF = ICHE ; . ' . OE is 11 to KH ; and since EP is bisected in K, OP is bi­ sected in I. L s of the four 6 s AFE, ODE, ABO , DBF (see fig. , Gor. 2, Prop. 1 2), then X, Y, Z, W are col­ linear. enote the line of collinearity of the feet of the 1.. s from P on the sides of the four 6 s . Join PX, PY, PZ, PW. Then, since L j oins the points of bisection of the sides of the � PXY, the line XY is 11 to L.

S from the extremi­ ties of t he base on the bisector of the exter nal vertieal L i s a circle, whose centre i s the middle point of the base, and whose radius = half sum of the si des. = Prop. -The three perpend�'cular8 to the 8�'de8 of a ir£angle at their m�'ddle point8 are conctttrrent. -Let the middle poin ts be D, E, F. Draw FG, E G J.. to AB, AC, A and let these J.. s meet in G ; join GD : the prop. will be proved by sho"\v ing that GD is J.. to B C . Join AG, BG, CG. 6 s AFG and BFG, since AF = FB, and FG -�--� common, and the L B D c AFG = BFG, AG i s = GB (i v.

If the two 0 8 touch, the square of their common tangent is equal to the rectangle contained by their diameters. Oor. �The square of the transverse common tan­ gent of the two 0 8 = AD . BC. Oor. L::B QR be the diameter of CQD, AB . QR = E P2 Dem. PB . QR = PQ,2, « (for . 1 ) ' . __ _____-I . AP . QR EP 2 PQ,2 ; A B . QR = EP2. = _ ( 6) therefore, by addition, Oor. B O OK Ill. 33 nate of a semicircle, the semicircle itself and the semi­ circles on the segments of the diameter, they will be equal to one another.

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

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