PROPOSITION XIX. THEGREA 213. If the sum of two arcs be less than a circumference the greater arc is subtended by the greater chord; and conversely, the greater chord subtends the greater arc. B C P In the circle ACP let the two arcs A B and BC together be less than the circumference, and let AB be the greater. We are to prove chord A B> chord BC. Draw A C. In the ▲ ABC ZC, measured by the greater arc A B, § 203 is greater than A, measured by the less arc B C. .. the side A B > the side BC, § 117 (in a ▲ the greater has the greater side opposite to it). CONVERSELY: If the chord AB be greater than the (in a ▲ the greater side has the greater ...arc A B, double the measure of the greater than the arc BC, double the measure of the less opposite to it). Q. E. D. 214. If the sum of two arcs be greater than a circumference, the greater arc is subtended by the less chord; and, conversely, the less chord subtends the greater arc. B A E In the circle BCE let the arcs AECB and BAEC together be greater than the circumference, and let arc AEC B be greater than arc BAEC. From the given arcs take the common arc A EC; we have left two arcs, CB and A B, less than a circumference, of which CB is the greater. .. chord C B > chord A B, § 213 (when the sum of two arcs is less than a circumference, the greater arc is subtended by the greater chord). .. the chord A B, which subtends the greater arc A EC B, is less than the chord BC, which subtends the less arc BAE C. CONVERSELY: If the chord A B be less than chord B C. ON CONSTRUCTIONS. PROPOSITION XXI. PROBLEM. 215. To find a point in a plane, having given its distances from two known points. Let A and B be the two known points; n the distance of the required point from A, o its distance from B. It is required to find a point at the given distances from A and B. From A as a centre, with a radius equal to n, describe an arc. From B as a centre, with a radius equal to o, describe an arc intersecting the former arc at C. C is the required point. Q. E. F. 216. COROLLARY 1. By continuing these arcs, another point below the points A and B will be found, which will fulfil the conditions. 217. COR. 2. When the sum of the given distances is equal to the distance between the two given points, then the two arcs described will be tangent to each other, and the point of tan Let the distance from A to B equal n + o. radius equal to n, describe an arc; A· and from B as a centre, with a radius equal to o, describe an arc. These arcs will touch each other at C, and will not intersect. n 0 .. C is the only point which can be found. .B 218. SCHOLIUM 1. The problem is impossible when the distance between the two known points is greater than the sum of the distances of the required point from the two given points. Let the distance from A to B be greater than n + o. 219. SCHO. 2. The problem is impossible when the distance between the two given points is less than the difference of the distances of the required point from the two given points. Let the distance from A to B be less than n From A as a centre, with a radius equal to n, describe a circle; 0. Let A B be the given straight line. It is required to bisect the line A B. From A and B as centres, with equal radii, describe arcs intersecting at C and E. Join CE. Then the line CE bisects A B. For, Cand E, being two points at equal distances from the extremities A and B, determine the position of a to the middle point of A B. PROPOSITION XXIII. PROBLEM. § 60 Q. E. F. 221. At a given point in a straight line, to erect a perpendicular to that line. R Let O be the given point in the straight line A B. It is required to erect a to the line A B at the point 0. From B and H as centres, with equal radii, describe two arcs intersecting at R. Then the line joining RO is the required. For, O and Rare two points at equal distances from B and H, and .. determine the position of a 1 to the line HB at its middle point 0. § 60 |