PROBLEMS FOR BOOK 1 57 Problems for Book 1 Exercise 39. In a triangle, the larger side corresponds to the smaller median1. (Consider the angle made by the third median with the third side.) A triangle with two equal median is isosceles. Exercise 40. Let us assume that a billiard ball which strikes a flat wall will bounce off in such a way that the two lines of the path followed by the ball (be- fore and after the collision) make equal angles with the wall. Consider n lines D1, D2, . . . , Dn in the plane, and points A, B on the same side of all of these lines. In what direction should a billiard ball be shot from A in order that it arrive at B after having bounced off each of the given lines successively? Show that the path followed by the ball in this case is the shortest broken line going from A to B and having successive vertices on the given lines.2 Special Case. The given lines are the four sides of rectangle, taken in their natural order the point B coincides with A and is inside the rectangle. Show that, in this case, the path traveled by the ball is equal to the sum of the diagonals of the rectangle. Exercise 41. The diagonals of the two rectangles of exercise 26 are situated on the same two lines, parallel to the sides of the given parallelogram (analogous to 54). One of these diagonals is half the difference, and the other half the sum, of the sides of the parallelogram. Exercise 42. In an isosceles triangle, the sum of the distances from a point on the base to the other sides is constant.— What happens if the point is taken on the extension of the base? In an equilateral triangle, the sum of the distances from a point inside the triangle to the three sides is constant. What happens when the point is outside the triangle? Exercise 43. In triangle ABC, we draw a perpendicular through the midpoint D of BC to the bisector of angle A. This line cuts off segments on the sides AB, AC equal to, respectively, 1 2 (AB + AC) and 1 2 (AB − AC). Exercise 44. Let ABCD, DEF G be two squares placed side by side, so that sides DC, DE have the same direction, and sides AD, DG are extensions of each other. On AD and on the extension of DC, we take two segments AH, CK equal to DG. Show that quadrilateral HBKF is also a square. Exercise 45. On the sides AB, AC of a triangle, and outside the triangle, we construct squares ABDE, ACGF , with D and F being the vertices opposite A. Show that: 1◦. EG is perpendicular to the median from A, and equal to twice this median 2◦. The fourth vertex I of the parallelogram with vertices EAG (with E and G opposite vertices) lies on the altitude from A in the original triangle 1 For similar statements concerning the altitudes of a triangle, see Exercises 19, 20, and for angle bisectors see exercises 362, 362b at the end of this volume. 2 In the case where there is only one line, the problem reduces to the subject of Exercises 13–14. Once these exercises are solved, one tries to find a way to use the solution for the case of one line to treat the case for two lines then to extend it to three lines, and so on.

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