CHAPTER XV. Circles In and About Triangles. 1. If the angle BAC, between two lines, be bisected, and, from any point D in it, perpendiculars DB, DC be drawn, these perpendiculars are evidently equal. If, then, a circle be described with centre D, and radius DB or DC, it will touch both the lines. Thus all circles touching both lines have their centres in the straight line which bisects the angle between the lines. Two lines make an angle of 120° with one another. Describe four circles, of different radii, touching both of them. Two lines make an angle of 80° with one another. Describe a circle of radius in. to touch both of them; also of radius 1 in. Two lines make an angle of 60° with one another. Describe a circle touching both of them; also a second circle touching the previous circle and the two lines. 2. We may describe a circle touching the three sides of a triangle as follows: Bisect the angles at B and C by the lines BD, CD. Then BD contains the centres of circles touching BA and BC; and CD contains the centres of circles touching CA and CB. Hence D is the centre of a circle which touches all three sides. DE, perpendicular to BC, is the radius of this circle. and radius DE, describe a circle. inscribed in the triangle ABC. B E Hence with centre D The utmost care is to be exercised in accurately bisecting the angles; otherwise it may be found that, when the circle is described, it cuts a side, or falls short of one. Inscribe a circle in the triangle whose sides are 75, 80 and 95 millimetres. Describe a circle to touch the other side of BC (any triangle ABC), and the sides AB and AC produced. The base of a triangle is 2 in., and the angles at the base are 40° and 110°. Inscribe a circle in it. Measure its radius. 3. We have already (Ch. X., 6), in effect, shown how to describe a circle about any triangle, i.e., to pass through the angular points of the triangle. Two sides, say AB and AC, are bisected, and DO, EO are drawn through the points of bisection B A E perpendicular to AB and AC, respectively. Then all points in DO are equally distant from A and B; and all points in EO are equally distant from A and C. Hence O is equally distant from A, B and C ; and if the sharp point of the compasses be placed at O, and the pencil end at A, or B, or C, and a circle be described, it will pass through A, B and C. Here again the greatest care must be exercised in bisecting the sides, and in drawing the perpendiculars at the points of bisection; otherwise the circle will pass through the angle on which the pencil end of the compasses was placed, but may not pass through the two other angles. Describe a circle about a triangle whose sides are 55, 70 and 90 millimetres. Measure its radius. The side of an equilateral triangle is 3 in.; describe a circle about it. Each of the equal sides of an isosceles triangle is 3 in., and the equal angles are each 75°. Describe a circle about it. Should the course contained in this book prove too long for a year's work, it is suggested that Chapters XVI., XVII. and XVIII. be omitted, valuable though they may be as affording exercises in accurate geometrical construction. Exercises. 1. Draw two lines making an angle of 50° with one another, and describe three circles touching both lines. Describe a 2. Two lines make an angle of 70° with one another. circle of radius 11⁄2 in. touching both of them. (Draw a perpendicular to either of the lines, of length 11⁄2 in., and through its end draw a line parallel to the line on which the perpendicular stands, producing this parallel until it meets the bisecting line.) 3. Two lines make an angle of 40° with one another. Describe a circle touching both of them; also a second circle touching the previous circle and the two lines. (At point where first circle cuts bisecting line, draw a line making an angle of 55° or 35° with it, according to cutting point selected.) 4. Describe a triangle with angles 30°, 60° and 90°, and hypotenuse 3 in., and in it inscribe a circle. 5. Describe a triangle with angles 30°, 60° and 90°, and hypotenuse 6 in., and in it inscribe a circle. Compare the length of the radius of this circle with length of the radius of circle in previous question. 6. Describe a triangle with sides 76, 68 and 44 millimetres, and in it inscribe a circle. 7. In the case of the triangle of the previous question, describe circles touching each side and the other two sides produced. 8. Having obtained the four circles of the two previous questions, through what points do the lines joining any two centres pass? What position does the line joining any two centres occupy with respect to the line joining the other two centres? Apply tests in both cases. 9. Two parallel lines are 1 in. apart, and a third line cuts them at an angle of 60°. Describe all the circles you can, each touching the three lines. What is the length of the radius? 10. In the previous question, what is the figure formed by joining the centres to the points where the parallels are cut by the third line? Apply test. 11. Is there any position which three lines can occupy, such that no circle can be described touching all? 12. Describe an equilateral triangle with side 2 in., and in it inscribe a circle. Express with exactness the radius of this circle. 13. Describe also a circle about the triangle of the previous question, and express with exactness its radius. 14. Construct a triangle with sides 40, 45 and 50 millimetres, and about it describe a circle. 15. Construct a triangle with sides 80, 90 and 100 millimetres, and about it describe a circle. Compare the length of radius of this circle with that of circle in previous question. 16. Is there any position which three points can occupy with respect to one another, such that a circle cannot be described to pass through all? 17. ABCD is a quadrilateral; A=85°, B=80°, C=95°; AB=60 and BC 80 millimetres. Construct the quadrilateral and describe a circle about it. 18. AB (=3 in.) and CD (=2 in.) are parallel and 1 in. apart. A line at right angles to one and through its bisection passes also through the bisection of the other. Describe a circle to pass through A, B, C, D. 19. A line AB is 3 in. long. Describe a circle of radius 3 in. to touch AB at A. Describe a second circle to touch the previous one and also AB at B. 20. From the fact that two tangents from the same point to a circle are equal, what relation can you establish between the sums of the opposite sides of a quadrilateral whose sides touch a circle? 21. Construct a quadrilateral whose sides are 40, 30, 50 and 60 millimetres, and inscribe a circle in it. |