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By John Casey

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

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The 6 ECK = DBFo (3). EK2 + FD2 = 5BC2. ) ; . 17 B O OK 1. B, and NI{ EK2 = AB2 + 4AC 2 . = 2AC ; In like manner FD2 = 4AB2 + AC� ; E K2 + FD2 = 5 (AB 2 + A. C 2 ) therefore = 5 B C2. (4) . The intercepts AQ, AR are equal . ( 5 ) . The lines OF, BK, AL are concurrent. SECTION 11, EXERCISES. 1 . The line which bisects the vertical L of an isosceles 6. bisects the base perpendicularly. 2. The diagonals of a quadrilateral whose four sides are equal bisect each other perpendicularly. 3 . If the line which bisects the vertical L of a 6.

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 .

We �·nsert a few here, confin�'ng ourselves to those that may oe proved by the First Book. ( 1 ) . The transverse lines AE, l3K are 1- to each other. For, in the 6 s ACE, BCK, ,vhich are in every G respect equal, the L EAC = B1(C, and the L AQO = KQC ; hence the angle AOQ = KCQ, and is . F. right angle. (2). 6 KCE = Dl3F . Delu. L EN . Now, the L ACN = BCE, each bein g a right angle ; . . L ;0 BC = CE ; hence (xxvi. ) EN = AB and CN = AC ; but AC = CK ; . . CN = CK, and the £:... ) ; but the 6 ENC = ABC ; hence the 6 ECK = ABC.

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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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