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 dis tance 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 ares, 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 tangency will be the point required. Let the distance from A to B equal n + o. From A as a centre, with a radius equal to n, describe an arc; A: and from B as a centre, with a radius equal to 0, describe an •B n arc. 0 These arcs will touch each other at C, and will not intersect. .. C is the only point which can be found. 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 + 0. Then from A as a centre, with a radius equal to 1, de- A• •B scribe an arc; n 0 and from Bas a centre, with a radius equal to o, describe an arc. These arcs will neithertouch nor intersect each other; hence they can have no point in common. 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 - 0. From A as a centre, with a radius equal to n, describe a circle ; and from B as a centre, with a radius equal to o, describe a circle. A• •B The circle described from B as a centre will fall wholly within the circle described from A as a centre; hence they can have no point in 0 n common. PROPOSITION XXII. PROBLEM. 220. To bisect a given straight line. c E From A and B as centres, with equal radii, describe arcs intersecting at C and E. Join CE. Then the line C E bisects A B. For, C and E, being two points at equal distances from the extremities A and B, determine the position of a I to the middle point of A B. § 60 Q. E. F. PROPOSITION XXIII. PROBLEM. 221. At a given point in a straight line, to erect a perpendicular to that line. R A -B H 0 Let O be the given point in the straight line A B. It is required to erect a I to the line A B at the point 0. Take O H OB. From B and H as centres, with equal radii, describe two ares intersecting at R. Then the line joining R O is the I required. For, O and R are two points at equal distances from B and H, and .. determine the position of a I to the line H B at its middle point o. $ 60 Q. E. F 222. From a point without a straight line, to let fall a perpendicular upon that line. Let A B be a given straight line, and C a given point without the line. It is required to let fall a I to the line A B from the point C. From C as a centre, with a radius sufficiently great, describe an arc cutting A B at the points H and K. From H and K as centres, with equal radii, describe two arcs intersecting Draw CO, and produce it to meet A B at m. C m is the I required. For, C and O, being two points at equal distances from H and K, determine the position of a I to the line HK at its middle point. $ 60 Q. E. F. PROPOSITION XXV. PROBLEM. 223. To construct an arc equal to a given arc whose centre is a given point. Let C be the centre of the given arc A B. It is required to construct an arc equal to arc A B. Draw C B, C A, and A B. From Cas a centre, with a radius equal to CB, describe an indefinite arc B' F. From B' as a centre, with a radius equal to chord A B, describe an arc intersecting the indefinite arc at A'. |