1031. Of all the polygons constructed with the same given sides, that is the maximum which can be inscribed in a circle.

GIVEN-a polygon P inscribed in a circle, and P', any other polygon constructed with the same sides and not inscriptible in a circle.

Upon the sides of the polygon P'' construct circular segments equal to those on the corresponding sides of P.

The whole figure S' thus formed has the same perimeter as the circle S.

1032. Of all isoperimetric polygons having the same number of sides the maximum is a regular polygon.

GIVEN-P the maximum of all the isoperimetric polygons of the same number of sides.

TO PROVE

that P is a regular polygon.

If two of its sides, as AB', B'C, were unequal, the isosceles triangle ABC, having the same perimeter as AB'C and a greater area, could be substituted for the triangle ABC. § 1025 This would increase the area of the whole polygon without changing the length of the perimeter or the number of its sides. Hence the sides of the maximum polygon must be all equal.

But the maximum of all polygons constructed with the same given sides must be inscriptible in a circle.

Therefore P is a regular polygon.

§ 1031 §§ 164, 469

Q. E. D.

PROPOSITION XVI. THEOREM

1033. Of all polygons having the same number of sides and the same area, the regular polygon has the minimum perimeter.

GIVEN-P, a regular polygon, and M, any other polygon of the same number of sides and area as P.

TO PROVE

the perimeter of P< perimeter of M.

Let N be a regular polygon having same perimeter and same number of sides as M.

Then area Marea N, or area P<area N.

§ 1032

But of two regular polygons of the same number of sides the one of less area has the less perimeter.

$$ 482, 483

Therefore the perimeter P is less than that of N, or less than that of M.

Q. E. D.

4

EXERCISES

BOOK I

PROBLEMS OF DEMONSTRATION

1. The bisector of an angle of a triangle is less than half the sum of the sides containing the angle.

2. The median drawn to any side of a triangle is less than half the sum of the other two sides, and greater than the excess of that half sum above half the third side.

3. The shortest of the medians of a triangle is the one drawn to the longest side.

4. The sum of the three medians of a triangle is less than the sum of the three sides, but greater than half their sum.

5. In any triangle the angle between the bisector of the angle opposite any side and the perpendicular from the opposite vertex on that side is equal to half the difference of the angles adjacent to that side.

6. LM and PR are two parallels which are cut obliquely by AB in the points A, B, and at right angles by AC in the points A, C; the line BED, which cuts AC in E and LM in D, is such that ED is equal to 2AB. Prove that the angle DBC is one-third the angle ABC.

7. The sum of the diagonals of a quadrilateral is less than the sum of the four lines joining any point other than the intersection of the diagonals to the four vertics.

8. The difference between the acute angles of a right triangle is equal to the angle between the median and the perpendicular drawn from the vertex of the right angle to the hypotenuse.

9. In a right triangle the bisector of the right angle also bisects the angle between the perpendicular and the median from the vertex of the right angle to the hypotenuse.

10. In the triangle formed by the bisectors of the exterior angles of a given triangle, each angle is one-half the supplement of the opposite angle in the given triangle.

11. A right triangle can be divided into two isosceles triangles.

12. A median of a triangle is greater than, equal to, or iess than half of the side which it bisects, according as the angle opposite that side is acute, right, or obtuse.

13. The point of intersection of the perpendiculars erected at the middle of each side of a triangle, the point of intersection of the three medians, and the point of intersection of the three perpendiculars from the vertices to the opposite sides are in a straight line; and the distance of the first point from the second is half the distance of the second from the third.

14. Find the locus of a point the sum or the difference of whose distances from two fixed straight lines is given.

15. On the side AB, produced if necessary, of a triangle ABC, AC' is taken equal to AC; similarly on AC, AB' is taken equal to AB, and the line B'C' drawn to cut BC in P. Prove that the line AP bisects the angle BAC.

16. The point of intersection of the straight lines which join the middle points of opposite sides of a quadrilateral is the middle point of the straight line joining the middle points of the diagonals.

17. The angle between the bisector of an angle of a triangle and the bisector of an exterior angle at another vertex is equal to half the third angle of the triangle.

18. If Land M are the middle points of the sides AB, CD of a parallelogram ABCD, the straight lines, DL, BM trisect the diagonal AC. 19. ABC is an equilateral triangle; BD and CD are the bisectors of the angles at B and C. Prove that lines through D parallel to the sides AB and AC trisect BC.

20. The angle between the bisectors (produced only to their point of intersection) of two adjacent angles of a quadrilateral is equal to half the sum of the two other angles of the quadrilateral. The acute angle between the bisectors of two opposite angles of a quadrilateral is equal to half the difference of the other angles.

21. The bisectors of the angles of a quadrilateral form a second quadrilateral of which the opposite angles are supplementary. When the first quadrilateral is a parallelogram, the second is a rectangle whose diagonals are parallel to the sides of the parallelogram and each equal to the difference of two adjacent sides of the parallelogram. When the first quadrilateral is a rectangle, the second is a square.

22. Two quadrilaterals are equal if an angle of the one is equal to an angle of the other, and the four sides of the one are respectively equal to the four similarly situated sides of the other.

23. If two polygons have the same number of sides and this number is odd, and if one polygon can be placed upon the other so that the middle points of the sides of the first fall upon the middle points of the sides of the second, the polygons are equal.

PROBLEMS OF CONSTRUCTION

24. Find a point in a straight line such that the sum of its distances from two fixed points on the same side of the straight line shall be the least possible.

25. Find a point in a straight line such that the difference of its distances from two fixed points on opposite sides of the line shall be the greatest possible.

26. Draw through a given point within a given angle a straight line such that the part intercepted between the sides of the angle shall be bisected by the given point.

27. Through a given point without a straight line to draw a straight line making a given angle with the given line.

28. Divide a rectangle 7 in. long and 3 in. broad into three figures which can be joined together so as to form a square.

BOOK II

PROBLEMS OF DEMONSTRATION

29. If a circle is circumscribed about an equilateral triangle and from any point in the circumference straight lines are drawn to the three vertices, one of these lines is equal to the sum of the other two.