30 II. ON TRIANGLES

Exercise 9. The sum of the diagonals of a

[convex5]

quadrilateral is between

the semi-perimeter and the whole perimeter.

Exercise 10. The intersection point of the diagonals of a

[convex6]

quadrilat-

eral is the point in the plane such that the sum of its distances to the four vertices

is as small as possible.

Exercise 11. A median of a triangle is smaller than half the sum of the sides

surrounding it, and greater than the diﬀerence between this sum and half of the

third side.

Exercise 12. The sum of the medians of a triangle is greater than its semi-

perimeter and less than its whole perimeter.

Exercise 13. On a given line, ﬁnd a point such that the sum of its distances

to two given points is as small as possible. Distinguish two cases, according to

whether the points are on the same side of the line or not. The second case can be

reduced to the ﬁrst (by reflecting part of the ﬁgure in the given line).

Exercise 14. (Billiard problem.) Given a line xy and two points A, B on the

same side of the line, ﬁnd a point M on this line such that AMx = BMy.

We obtain the same point as in the preceding problem.

Exercise 15. On a given line, ﬁnd a point with the property that the diﬀerence

of its distances to two given points is as large as possible. Distinguish two cases,

according to whether the points are on the same side of the line or not.

5This

condition is not in the original text. See solution. –transl.

6See

the previous footnote. –transl.