Measure the numbers of millimetres in each of the lines EA, EB, EC in the first figure, and exainine whether the product of EA and EB is approximately equal to the square of EC. Describe other circles, draw to each a secant and a tangent from the same point, and repeat in the case of each the construction of the second figure. Repeat also the measurements and multiplications. The result of our observations may be stated as follows: If from any point without a circle two straight lines be drawn, one a secant and the other a tangent, then the rectangle contained by the secant and the part of it without the circle is equal to the square on the tangent. If another secant EDF be drawn, since the rectangle con E tained by EA and EB is equal to the square EC, and the rectangle contained by ED and EF is equal to the square EC, therefore the rectangle contained by EA and EB is equal to the rectangle contained by ED and EF. The segments of one chord are 3, 4, and of another 2, 6 quarters of an inch, the chords making an angle of 30° with one another, describe the circle through the extremities of the chords. If the segments of another line through the intersection of the chords be 14 and 8 quarters of an inch, do the ends of this necessarily on on F rest on the circle ? Place the line that its ends may so rest. The tangent to a circle is 60 millimetres; a secant is 90, and the part of it without the circle 40 millimetres. These lines make an angle of 60° with one another. Describe the circle. Exercises. 1. Two lines AB, CD intersect in E. AE=30, EB=40, CE=20, ED=60 millimetres, so that AE. EB=CE. ED. Show that a circle can be described to pass through the four points A, C, B, D, i.e., that a circle through A, D, B, say, also passes through C. 2. Two lines, AB, CD cut one another in E. AE=13, EB=2, CE=3, ED=1 in., so that AE. EB=CE. ED. Describe a circle to pass through A, C, B, D. 3. Describe a circle of radius 2 in. Draw a diameter AB. Take in it a point C at distance 1 in. from centre, and draw chord DCE perpendicular to AB. By construction, as in text, show that rectangle AC.CB is equal to square on CD. It may also be shown that CD= 13 in. by proving it equal to the altitude of an equilateral triangle whose side is 2 in. 4. Describe a circle of radius in 23 in. Draw a diameter AB. In it take a point C at distance 1} in. from centre, and draw chord DCE perpendicular to AB. What should be the length of CD? Measure it. 5. Two lines intersect at an angle of 30°. The segments of one 2 in. and i in., of the other, both 1 in. Describe a circle to pass through the ends of the lines. With what inclination of the lines to one another would the longer line become a diameter? 6. Describe a circle of radius 3 in. In it place a chord of length 4 in., and take in the chord a point at distance 1 in. from an end. Through this point draw another chord whose segments shall be 13 in. and 2 in. 7. Describe a circle of radius 70 millimetres. In it place a chord of length 90 millimetres, and take a point in the chord at distance a 40 millimetres from an end. Through this point draw two chords whose segments shall be 20 and 100 millimetres. 8. On a line take lengths, AB, AC, of 27 and 48 millimetres, in the same direction. Draw a line AD of 36 millimetres, making an angle of 45° with AC. Describe a circle through B, C, D. What is AD with respect to this circle? 9. Same problem as previous, but with AB=36, AC=64, AD=48 millimetres, and angle between AC, AD, 60°. Describe a circle through B, C and D. What position does AD occupy with respect to it? 10. AB, AC, measured along the same line, in the same direction, are 36 and 64 millimetres ; and AD another line through A is 48 millimetres. Place AD so that the circle through B, C and D may have its centre in AC. 11. AB, AC measured along the same line, in the same direction, are 18 and 72 millimetres. Describe a number of circles through B and C, and from A draw a tangent to each. Measure the lengths of these tangents. What relation between the lengths and why? 12. Two lines AB, AC of length 13 in., both touch the same circle at B and C, and make an angle of 60° with one another. Construct the circle. What is its radius? 13. AB, AC measured along the same line in the same direction are 48 and 108 millimetres. Describe a circle on BC as diameter, and draw a line ADE cutting the circle in D and E, such that AD=54 millimetres. What is the length of AE? Draw a tangent to the circle from A. What is its length ? 14. Describe two circles of radii 1 and 2 inches respectively, intersecting in A and B. Draw a straight line through A and B, and from any point in it, draw a tangent to each circle. Measure the tangents. What relation between their lengths? Give reason. 15. Describe two circles of radii 25 and 70 millimetres, intersecting in A and B. Draw a straight line through A and B, and from any point in it draw a tangent to each circle. Measure the tangents. What relation between their lengths ? Give reason. 16. Describe three circles of radii 2, 3 and 34 inches, so that each intersects the other two. Through each pair of points of intersection draw straight lines. These three lines should pass through the same point. 17. If the tangents to two intersecting circles from any point be equal, that point must lie on the line joining the points of intersection of the circles. 18. The common chord of two intersecting circles on being produced, cuts a line that touches both circles. Show that the tangent line must be bisected. 19. ABC is a triangle right-angled at C, and from Ca perpendicular CD is drawn to AB. By describing a circle about ABC, show that the rectangle AD.DB is equal to the square on CD. 20. ABC is a triangle right-angled at C, and from C a perpendicular CD is drawn to AB. By describing a circle about the triangle CDB, show that the rectangle AD. AB is equal to the square on AC. 21. In the previous question, describe a circle about the triangle ACD, and show that the rectangle BA.BD is equal to the square on BC. 22. The sides of a triangle are 3, 4, 5, and a perpendicular is dropped from the right angle in the hypotenuse. Find the lengths of the segments of the hypotenuse on each side of the perpendicular, and also the length of the perpendicular. Triangles In and About Circles. 1. A triangle is said to be inscribed in a circle when the three angular points of the triangle rest on the circumference of the circle. We evidently cannot in general construct in a circle of given size a triangle equal to a given triangle. In a small circle we could not place a large triangle. Indeed we have seen (Ch. X., 6) that there is but one circle which can be made to fit round a triangle of given size. We can, however, always inscribe in any circle a triangle equiangular to another triangle, i.e., a triangle with its angles of given size, their sum of course being 180°. Thus let it be required to construct in a given circle a triangle whose angles shall be 30°, 70, 80o. Using the protractor, adjust the bevel to an angle equal to any one of these, say, 30°. Place the angle of the bevel at any point C on the circumference, and with a needle mark the points, A and B, where the legs of the bevel cross the circumference. We have thus a segment ACB containing an angle of 30°, and all angles in the segment ACB are angles of 30°. With the protractor at A make the angle BAD of 80°. Join BD. Then ADB is an angle of 30°. Hence the remaining angle ABD is 70 Of course the angle of 30° at C may be constructed |