Problem 128. To inscribe five equal circles in a given circle A. 1. Divide the circumference into five equal parts, as in the case of inscribing a pentagon (Pr. 63). 2. From the centre A, draw lines through two divisions, as B and C, and produce them. 3. Bisect the angle BAC (Pr. 4), and draw AD, touching the circumference of the given circle in D. 4. At D, draw a tangent to the circle (Pr. 54), cutting AB and AC produced, and completing the triangle EAF. 5. Inscribe a circle in this triangle (Pr. 107), having its centre at G. 6. From A, with AG as radius, inscribe the circle GHKLM, cutting AE and AF in N and O. 7. From A, with the line NO as radius, cut the circum ference of the inner circle in H, K, L, M. 8. From those points, with radius DG, describe the remaining four circles within the given circle A, to complete the figure. 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 X, 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. 1. Draw a diameter BC, and from the radius of the circle, divide the piu B, with the into six 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 (6.) 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 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 C... 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. |