Euclid's Elements: Books I, II, III . Let A ABC be an obtuse-angled A, having L ACBobtuse. From A draw AD J- EC is required to prove sq. on AB = sq. on BC + sq. on CA + twice rect. Proof. Now sq. on BD = sq. on BC + sq. on CD + twice rect. //. 4. To each add sq. on sq. on BD + sq. on DA = sq. on BC + sq. on CD+ sq. on DA + twice rect. . L ADC is a right angle,.. sq. on BD + sq. on DA = sq. on AB,and sq. on CD + sq. on DA = sq. on CA. /. sq. on AB = sq. on BC + sq. on CA+ twice rect. Book II.] EUCLID S ELEMENTS. .139 AN ALTERNATIVE PROOF OF
Euclid's Elements: Books I, II, III . Let A ABC be an obtuse-angled A, having L ACBobtuse. From A draw AD J- EC is required to prove sq. on AB = sq. on BC + sq. on CA + twice rect. Proof. Now sq. on BD = sq. on BC + sq. on CD + twice rect. //. 4. To each add sq. on sq. on BD + sq. on DA = sq. on BC + sq. on CD+ sq. on DA + twice rect. . L ADC is a right angle,.. sq. on BD + sq. on DA = sq. on AB,and sq. on CD + sq. on DA = sq. on CA. /. sq. on AB = sq. on BC + sq. on CA+ twice rect. Book II.] EUCLID S ELEMENTS. .139 AN ALTERNATIVE PROOF OF PROPOSITION 12. M. Let ARC be an obtuse-ang-led A, having L ACB AX J. BC produced, and BQ i. AC is required to prove sq. on AB = sq. on AC + sq. on BC + twice rect. (or twice rect. ).Construction. Describe squares on the sides of the triang^Ie andcomplete the rectangles CM, CR ± CD, AF, AG, Now A CBD = A rect. BR = rect. similarly rect. AR = rect. , A CBH = A rect. CM = rect, AB = sq. on AC + sq. on BC + twice rect. CM • ^q- = sq. on AC 4- sq. on BC + twice rect. ?VTr:^.^- ^ 140 EUCLID S ELEMENTS. [Book II. PROPOSITION 13. Theorem. In evepy triangle the square on the side subtendingr anacute ang-le is less than the sum of the squareson the sides containing- that angrle, by twice thepectangle contained by either of these sides, and theline intercepted between the perpendicular let fallon it from the opposite angle and the acute angle.
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