By the use of the method and notation of § 354 it is found that This formula holds for the bisector of the angle C. Analogous formulas hold for the bisectors of angles A and B. PROPOSITION XXIV. THEOREM 427. In any triangle the product of two sides is equal to the product of the altitude upon the third side by the diameter of the circumscribed circle. Given d, the diameter CE of the circle circumscribed about the triangle ABC, and the altitude h upon the side c. Suggestion. Draw BE and compare & ACD and ECB. 428. From the above relation, and from the formula (§ 354) for the altitude upon any side of a triangle, the formula for the length of the radius of the circumscribed circle may be derived as follows: PROPOSITION XXV. THEOREM 429. In any triangle the area is equal to the product of half the perimeter and the radius of the inscribed circle. Given the triangle ABC, and the radius r of the inscribed circle. To prove that area △ABC = (a + b + c)r. (To be completed.) 430. From the above relation, and from the formula for the area of a triangle (§ 354), may be derived a formula for the length of the radius of the circle inscribed in any triangle, namely, 1. The sides of a triangle are 13, 14, 15. Find the projection of 15 upon 14, and find the altitude upon 14. 2. In the following table ascertain whether the triangles indicated 3. The sides of a triangle are 6, 7, and 9. Find the projections of 9 and 7 upon 6. 4. Two sides of a triangle are 10 and 12, and they inclose an angle of 45°. Find the projection of 10 on 12, and of 12 on 10. Find the altitude on 12, and the area of the triangle. 5. Two sides of a triangle are 7 and 8, and they inclose an angle of 60°. Find the projection of the side 8 on the side 7, and the third side of the triangle. 6. Two sides of a triangle are 10 and 12, and they inclose an angle of 30°. Find the projection of the side 10 on the side 12, the altitude, and the area of the triangle. 7. Two sides of a triangle are 8 and 9, and the angle between them is 120°. Find the third side and the area. 8. Two sides of a triangle are 18 and 30, the included angle is acute, and the projection of the first upon the second is 12. Find the third side. 9. One side of an acute triangle is 20, and the projection of another side upon it is 10. What is known about this triangle? Is the triangle determined definitely? 10. One side of an acute triangle is 8, and its projection on another side is 4. What is known about this triangle? Is the triangle determined definitely? 11. The sides of a triangle are 9, 12, and 15. Find the three altitudes. 12. The sides of a triangle are 5, 9, and 10. What kind of triangle is it? Find the three altitudes. 13. The sides of a triangle are 14, 16, 18. Find the length of the median on the side 16. 14. The sides of a triangle are 9, 10, and 11. Find the length of the median on the side 9. 15. The sides of a triangle are 14, 48, and 50. Find the area, the altitude on the side 50, and the radius of the circumscribed circle. 16. The legs of a right triangle are 21 and 28. Find the segments of the hypotenuse made by the altitude upon it. 17. Find the altitude on the hypotenuse, and the median on the hypotenuse of the right triangle mentioned in Ex. 16. 18. The sides of a triangle are 8, 26, and 30. Find the radii of the circumscribed and inscribed circles. 19. The sides of a triangle are 6, 7, and 8. Find the length of the bisector of the angle opposite the side 7, terminating in this side. EXERCISES THEOREMS AND LOCUS PROBLEMS 1. The difference of the squares of two sides of a triangle is equal to the difference of the squares of the segments made by the altitude upon the third side. 2. The sum of the squares of the four sides of a parallelogram is equal to the sum of the squares of the diagonals (§§ 204, 422). 3. The sum of the squares of the medians of a triangle is equal to three fourths the sum of the squares of the three sides (§ 422). 4. The sum of the squares of the four sides of any quadrilateral is equal to the sum of the squares of the diagonals increased by four times the square of the line joining the mid-points of the diagonals. Suggestion. Join the mid-point of one diagonal to the extremities of the other, and apply § 422. 5. If three perpendiculars upon the sides of a triangle meet in a point, the sum of the squares of one set of alternate segments of the sides is equal to the sum of the squares of the other set (§ 344). 6. Find the locus of points whose distances from two fixed parallel lines are in a given ratio. 7. Through a point A on a circle chords are drawn. On each one of these chords a point is taken one third the distance from A to the end of the chord. Find the locus of these points. 8. Given the base of a triangle in magnitude and position and the sum of the squares of the other two sides. Find the locus of the vertex (§ 422). 9. Given the base of a triangle in magnitude and position and the difference of the squares of the other two sides. Find the locus of the vertex (§ 423). 10. Plot the locus of a point if the product of its distances from two perpendicular lines is constant (§ 298). 11. The vertex A of a rectangle ABCD is fixed, and the directions of the sides AB and AD also are fixed. Plot the locus of the vertex C if the area of the rectangle is constant. (See Ex. 10.) BOOK V REGULAR POLYGONS AND CIRCLES CONSTRUCTION OF REGULAR POLYGONS 431. Definitions. A polygon that is both equiangular and equilateral is called a regular polygon (§§ 57, 95). A polygon whose sides are chords of a circle is called an inscribed polygon (§ 255). A polygon whose sides are tangent to a circle is called a circumscribed polygon (§ 277). EXERCISES 1. If a series of equal chords are laid off in succession on a circle, what relation exists between the arcs of those chords? between the central angles of those arcs? 2. What relation exists between the angles formed by the successive chords? Give a reason for your answer. A B E D C 3. Suppose that an inscribed polygon is formed of equal chords of a circle. Why would such a polygon be regular? 4. How many degrees in the central angle of a regular inscribed polygon of 4 (5, 6, 8, 10, 12, 15, 16) sides? State a general formula for the number of degrees in the central angle of a regular inscribed polygon of n sides. 5. Find the number of degrees in each interior angle of each of the polygons mentioned in the preceding exercise. What relation exists between the central angle and the interior angle of a regular inscribed polygon? Give proof. 6. Which of the angles referred to in Ex. 4 can be constructed geometrically by methods already shown (§§ 156, 158, 188)? Tabulate the results of the last three exercises. |