The elements of Euclid for the use of schools and colleges : comprising the first two books and portions of the eleventh and twelfth books; with notes and exercises . on ^i^ describe the squareAFGH. [1. 46. AB shall be divided at H sothat the rectangle AB, BH isequal to the square on AH. Produce GH to , because the straight lineAG \& bisected at E, and pro-duced to F, the rectangle CF, FA, together -^ith thesquare on AE, is equal to the square on EF. [II. 6. But EF is equal to EB. [Covstntdion. Therefore the rectangle CF, FA, together with the squareon AE, is equal to the square on
The elements of Euclid for the use of schools and colleges : comprising the first two books and portions of the eleventh and twelfth books; with notes and exercises . on ^i^ describe the squareAFGH. [1. 46. AB shall be divided at H sothat the rectangle AB, BH isequal to the square on AH. Produce GH to , because the straight lineAG \& bisected at E, and pro-duced to F, the rectangle CF, FA, together -^ith thesquare on AE, is equal to the square on EF. [II. 6. But EF is equal to EB. [Covstntdion. Therefore the rectangle CF, FA, together with the squareon AE, is equal to the square on the square on EB is equal to the squares on AE, AB,because the angle EAB is a right angle. [I. 47. Therefore the rectangle CF, FA, together with the squareon AE, is equal to the squares on AE, away the square on AE, which is common to both jtherefore the remainder, the rectangle CF, FA, is equal tothe square on AB. {Axiovi 3. But the figure FK is the rectangle contained by CF, FA^for FG is equal to FA ;and AD is the square on ^^B ;therefore F^is equal to AD. Take away the common part AK, and the remainder FHis equal to the remainder HD. {Axiom BOOK II. 11, 12. 67 But HD is the rectangle contained by AB, BR^ for AB isequal to BD; and FH is the square on AH; therefore the rectangle ^^,5^is equal to the square on AH, Wherefore the straight line AB is divided at H, so that the rectangle A B, BH is equcd to the square on AH. PROPOSITION 12. THEOREM. In obtuse-angled triangles, if a perpendicular he draicnfrom either of the acute angles to the opposite side pro-duced, the square on the side subtending the obtuse angle isgreater than the squares on the sides containing the obtuseangle, by twice the rectangle contained by the side onichich, when produced, the perpendicular falls, and thestraight line intercepted without the triangle, between thepetyendicular and the obtuse angle. Let ABC be an obtuse-angled triangle, having theobtuse angle ACB, and from the point A
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