Plane and solid geometry . angle ^if+ rectangle i>l><)= square CH -f square BF,19. .. square JjD=o= square CH+ square BF, Reasons 11. §489. 12. §235. 13. §489. 14. Arg. 9. 15. § 54, 1. 16. § 54, 7 a. 17. By steps simi- lar to 5-16. 18. §54,2. 19. §309. 508. Cor. I. The square described on either side of aright triangle is equivalent to the square described onthe hypotenuse ininus the square described on tlie otherside. 509. Cor. II. If similar polygons are described on thethree sides of a right triangle as hoinologous sides, thepolygon described on the hypotenuse is equivalent
Plane and solid geometry . angle ^if+ rectangle i>l><)= square CH -f square BF,19. .. square JjD=o= square CH+ square BF, Reasons 11. §489. 12. §235. 13. §489. 14. Arg. 9. 15. § 54, 1. 16. § 54, 7 a. 17. By steps simi- lar to 5-16. 18. §54,2. 19. §309. 508. Cor. I. The square described on either side of aright triangle is equivalent to the square described onthe hypotenuse ininus the square described on tlie otherside. 509. Cor. II. If similar polygons are described on thethree sides of a right triangle as hoinologous sides, thepolygon described on the hypotenuse is equivalent tothe sum of the polygons described on the other two sides. Given rt. A ABC, right-angled at C, and let P, Q, and i? be ^polygons described on a, 6, and c, respectively, as homol. prove i? =0= P + Q. Argument Only 1. PR a? 2. QR 3. • • R 0^ + 62 :1. 4. •• R<^P+Q. , Ex. 883. The square on the hypotenuse of an isosceles right triangleis equivalent to four times the triangle. 234 PLANE GEOMETRY. Pythagoras 510. Historical Note. Prop. XII is usually known as the Pythago-rean Proposition, because it was discovered by Pythagoras. The proofgiven here is that of Euclid (about 300 ). Pythagoras (569-500 ),one of the most famous mathe-maticians of antiquity, wasbom at Samos. He spent hisearly years of manhood study-ing under Thales and traveledin Asia Elinor and Egypt andprobably also in Babylon andIndia. He returned to Samoswhere he established a schoolthat was not a great he went to Crotona inSouthern Italy and theregained many adherents. Heformed, with his closest fol-lowers, a secret society, themembers of which possessedall things in common. They used as their badge the five-pointed star or pentagram which they knewhow to construct and which they considered symbolical of health. Theyate simple food and practiced severe discipline, having obedience, temper-ance, and purity as their ideals. The brotherhood regarded their leaderwith
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