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ON CONSTRUCTIONS.

PROPOSITION XXI. PROBLEM.

215. To find a point in a plane, having given its distances from two known points.

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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 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 o, describe an

arc.

These arcs will touch each

other at C, and will not intersect.

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.. 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 + o.

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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 no.

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.

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

common.

0

n

A B

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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.

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Q. E. F.

221. At a given point in a straight line, to erect a perpendicular to that line.

R

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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.

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From B and H as centres, with equal radii, describe two arcs intersecting at R.

Then the line joining RO is the required.

For, and R are two points at equal distances from B and H, and .. determine the position of a to the line HB at its

middle point Ọ.

§ 60

Q. E. F.

PROPOSITION XXIV. PROBLEM.

222. From a point without a straight line, to let fall a perpendicular upon that line.

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Let A B be a given straight line, and C a given point without the line.

It is required to let fall a 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 at 0.

Draw CO,

and produce it to meet A B at m.

Cm is the required.

For, C and O, being two points at equal distances from H and K, determine the position of a to the line HK at its middle point.

$ 60

Q. E. F.

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223. To construct an arc equal to a given arc whose centre is a given point.

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Let C be the centre of the given arc A B.

It is required to construct an arc equal to arc A B.

Draw CB, CA, 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'.

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