Problem 127.

To inscribe six equal circles within a given equilateral triangle. ABC.

1. Draw the lines BD, CE, and AG, bisecting the angles and sides of the given triangle, and cutting each other in 0.

2. Bisect the angle OBG (Pr. 4), and the point F, where the line of bisection cuts AG, will be the centre of one of the isosceles triangles into which the equilateral triangle has been divided.

3. Through F, draw HK parallel to BC (Pr. 9); and from H and K, draw HL and KL parallel to AB and AC, and cutting CE and BD in M and N.

4. From points F, H, K, L, M, N, with radius FG, describe the six required circles within the given equilateral triangle ABC.

Problem 128.

To inscribe seven equal circles in a given circle A.

1. Draw a diameter BC, and from the point B, with the radius of the circle, divide the circumference into six equal parts, in D, E, C, &c., and draw the radii.

H

2. Divide one of the radii, as AB, into three equal parts, in the points H, K.

3. From A, with radius AH, describe the central circle.

4. From A, with radius AK, describe a circle which, cutting the radii, will give the points L, M, N, O, P.

5. From points K, L, M, N, O, P, with radius AH, describe the six circles, which, with the central circle, constitute the seven required circles within the given circle A.

SECTION VIII.—DESCRIBED FIGURES.

DEFINITIONS.

1. Described figures. Described figures are either rectilineal or circular.

(a.) A rectilineal figure is said to be described about another rectilineal figure, when all the sides of the circumscribed figure pass through the angular points of the figure about which it is described. Ex. ABCD

A

B

(b.) A rectilineal figure is said to be described about a circle, when each side of the circumscribed figure touches the circumference of the circle.

Ex. ABCD

NOTE.-A circle is said to be described about a rectilineal figure, when the circumference of the circle passes through all the angular points of the figure about which it is described. Ex. A

A

Problem 129.

To describe an equilateral triangle about a given square ABCD.

1. From points A and B, with AC as radius, describe arcs cutting each other in G.

H

B

A

2. From G as centre, with the same radius, cut these arcs in E and F.

3. Join EA and FB, and produce them to meet in H.

4. Produce CD until it cuts the lines HE and HF produced in K and L; then HKL is the required equilateral triangle described about the given square ABCD.

Problem 130.

To describe an equilateral triangle about a given circle A.

1. Draw a diameter BC.

B

E

2. From B, with radius BA, cut the circumference in D and E.

3. From D, E, and C as centres, with DE as radius, describe arcs intercepting in G, F, and H.

4. Join GF, FH, and HG; then FGH is the required equilateral triangle described about the given circle A.

Problem 131.

To describe a triangle about a given circle O, having angles equal to those of a given triangle ABC.

1. Produce any side of the triangle, as BC, both ways to D and E.

2. Draw any radius OF, and draw GH as a tangent through

F (Pr. 54).