Problem 115.

To inscribe four equal semicircles, having their diameters adjacent, 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.

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, A 2, &c. In each of these five sectors a circle is to be inscribed.

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 produced 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.

4. From 4, mark off on each alternate diameter, the distances 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 O of the triangle ABC by bisecting two angles B and C (Pr. 22), and from O, 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).

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.

E

1. Find the centre O of the triangle ABC by bisecting the angles at B and C (Pr. 22), and draw 40.

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.

M

EX

B

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).