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Problem 115.

To inscribe four equal semicircles, having their diameters ad jacent, within a given square ABCD, each touching one side of the square.

1. Draw the diagonals AD and BC, also the diameters EF

and GH,

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2. Draw the remaining diagonals of each of the smaller

squares-viz., GE, EH, HF, and FG, intercepting the

former diagonals in K, L, M, and N. 3. Join these points by lines KL, LM, MN, NK, which are

the diameters on which to describe the four required semicircles within the given square ABCD.

Problem 116.

To inscribe any number of equal circles in this case five) in a given circle A.

General Method. 1. Divide the circumference into the same number of equal

parts as there are required circles to be inscribed in

this case five), and draw radii to each point of division, as A 1, X 2, &c. In each of these five sectors a circle is to be inscribed.

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2. Bisect one of them, as 4 A 3, by radius AB (Pr. 4), and

draw a tangent at B (Pr. 54), cutting A 4 and A 3 pro

duced in C and D. 3. Find E, the centre of the triangle ACD (Pr. 22), and

inscribe a circle in it. 4. From A as centre, with AE as radius, describe the inner

circle. 5. On this circle, mark off from E, the centres for the four

other circles, as F, G, H, K; and inscribe them. There will then be five equal circles, inscribed in the given circle A.

Problem 117.

To inscribe any number of equal semicircles in this case six) in a given circle A.

1. Draw the diameters EB and CD at right angles to each

other. 2. Divide the circumference into twice as many equal parts

as there are to be semicircles (in this case twelve equal parts) by trisecting each right angle (Pr. 5).

3. Draw the diameters FG, HK, LM, NO; and no matter

how many semicircles are required, join EC, and where it cuts the next diameter, as FG, we obtain a point P, which is the extremity of the diameter of one of the semicircles.

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4. From A, mark off on each alternate diameter, the dis

tances AQ, AR, AS, AT, AU, equal to AP. 5. Join PQ, QR, RS, ST, TU, and UP. These are the

diameters on which to inscribe the required semicircles in the given circle A.

Problem 118.

To inscribe three circles in any given triangle ABC, each touching two others, and two sides of the triangle.

1. Find the centre 0 of the triangle ABC by bisecting two

angles B and C (Pr. 22), and from 0, draw a perpendicular to each side of the triangle (Pr. 3) meeting AB

in D, BC in E, and AC in F. 2. Bisect two adjacent angles of each of three quadrilaterals

thus formed, the bisecting lines meeting in G, H, K, the centres of the three required circles, the radius of each circle being found by drawing a perpendicular from

each centre to one side of each quadrilateral, as GL to BE (Pr. 3).

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3. Inscribe the required circles G, H, and K in the given

triangle ABC

Problem 119.

To inscribe three circles in any given triangle ABC, each touching the other two, and one side of the triangle.

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1. Find the centre 0 of the triangle ABC by bisecting the

angles at B and C (Pr. 22), and draw AO. 2. Inscribe a circle in each of the triangles thus formed, by

bisecting two angles of each, the bisecting lines meeting

in D, E, F, the radius of each circle being found by drawing a perpendicular from the centre to one of the

sides of each triangle, as FG to BC (Pr. 22). 3. Inscribe the required circles D, E, and F in the given

triangle ABC

Problem 120. To inscribe three equal circles within a given equilateral triangle ABC, touching each other, and two sides of the triangle.

1. From A and B, and A and C, describe arcs intersecting

in D and E.

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2. Draw the lines BE and CD, cutting the centre of the

triangle in F. 3. Draw the line AG, and bisect the angle FGC (Pr. 4),

the line of bisection cutting CD in H. 4. From F, set off the distance FH on the lines AG and

BE, in the points K and L; then H, K, L will be the

three centres of the required circles. 5. Draw the line LH, and. HM will be the radius of the

required circles, to be inscribed in the given equilateral triangle ABC.

Problem 121.
To inscribe three equal circles in a given circle 0.

1. At any point A, draw a tangent AB (Pr. 54), and AC at

right angles to it (Pr. 2).

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