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By John Casey

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A A�----�--� B 22 A SEQUEL TO EUCLID . 1A 02 ]3 02 = = EO. AD2 + D 0 2 + 2AD . DE ; BD2 + D 02 - 2DB . D E . Hence, by addition, since (xii. ) (xiii. A 02 + ]3 02 = 2AD2 + 2D 02. This is a simple case of a very general Prop . , which we shall prove, on the properties of the centre of mean position for any system of points and any syst� m of multiples. The Props. ix. and x. of the Second Book are particular cases of this Prop . , viz . , when the point C i s in the line AB or the line AB produce d .

AP2 + "b . Bp z + c . Cp2 + a '. DP2, &c. , exceeds a . A02 by (a + + b . B02 b + c + + c. C02 + it . ) OP2. The foregoing proof may evidently be applied to this Proposition. The following is another proof from Townsend's Modern Geometry :From the points A, B, 0, D, &c. , let fall 1. , on the line OP ; then it is easy to see that 0 is the centre of mean position for the points A', R', C', D', and the system of multiples a, "b, c, a, &c. Now we have by Props . , Book 11. APZ = A02 + BP2 = B02 + cpz = C02 + DP2 = D02 + Op2 + Op2 + Op2 + Op2 + therefore, multiplying by a, "b, remembering that a .

A�B ; hence the base FG is greater than th� base AB . • e , Prop . -If 00, OD he any two h'n e8, AB any arc of a c�'rcle, or of any other curve concave to 0 ; then, of all the tangents wh�'ch can be drawn to AB, that whose �'n ter­ oept is bisected at the point of contact cuts off the minimum tr�·angle. -Let CD be bisected at P, and let EF, be any other tangent, Then through P draw GH 11 to EF ; then, since CD is bisected in P, the f::J. han the f::J. cut off by GH (I. 1 9) ; but the f::J. cut off by GH is less than the f::J.

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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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