53. Two oblique lines drawn from a point in a perpendicular, cutting off' equal distances from the foot of the perpendicular, are equal. Let FC be the perpendicular, and C A and C O two oblique lines cutting off equal distances from F. Fold over C FA, on C F as an axis, until it comes into the plane of C FO. F A will take the direction of FO, Point A will fall upon point 0, (FA = F 0, by hyp.). .. line C A = line CO, § 18 Q. E. D. PROPOSITION V. THEOREM. 54. The sum of two lines drawn from a point to the extremities of a straight line is greater than the sum of two other lines similarly drawn, but includeil by them. Let C A and C B be two lines drawn from the point C to the extremities of the straight line A B. . Let O A and O B be two lines similarly drawn, but included by C A and C B. We'are to prove CA + C B >0 A + 0 B. § 18 § 18 Produce A 0 to meet the line C B at E. (a straight line is the shortest distunce between two points), BE+O E > BO. and take away 0 E from each side of the inequality. Q. E. D. PROPOSITION VE. THEOREM. 55. Of two oblique lines drawn from the same point in a perpendicular, cutting off unequal distances from the foot of the perpendicular, the more remote is the greater. Let C F be perpendicular to A B, and C K and C H two oblique lines cutting off unequal distances from F. Draw E K and E H. $ 53 (two oblique lines drawn from the same point in a I, cutting off equal dis tances from the foot of the I, are equal). But CH + HE > C K + KE, $ 54 (The sum of two oblique lines drawn from a point to the extremities of a straight line is greater than the sum of two other lines similarly drawn, ..2 C H > 2 C K; . Q. E. D. 56. COROLLARY. Only two equal straight lines can be drawn from a point to a straight line; and of two unequal lines, the greater cuts off the greater distance from the foot of the perpendicular. PROPOSITION VII. THEOREM. 57. Two equal oblique lines, drawn from the same point in a perpendicular, cut off equal distances from the foot of the perpendicular. Let C F be the perpendicular, and C E and C K be two equal oblique lines drawn from the point C. Fold over CFA on C F as an axis, until it comes into the plane cf C FB. The line FE will take the direction FK, (2 CFE= 2 C F K, cach being a rt. Z). otherwise one of these oblique lines must be more remote from the I, and .. greater than the other; which is contrary to the hypothesis. § 55 .. FE= FK. Q. E. D. PROPOSITION VIII. THEOREM. 58. If at the middle point of a straight line a perpendicular be erected, I. Any point in the perpendicular is at equal distances from the extremities of the straight line. II. Any point without the perpendicular is at unequal distances from the extremities of the straight line. Let PR be a perpendicular erected at the middle of the straight line A B, 0 any point in PR, and C any point without P R. Draw 0 A and O B. We are to prove OA= 0 B. PA = PB, § 53 (two oblique lines drawn from the same point in a I, cutting off equal dis tances from the foot of the I, are equal). One of these lines, as C A, will intersect the I. From D, the point of intersection, draw D B. |