(20.) Show that if R is the radius of a sphere and h the altitude of a spherical segment of one base, the volume of the spherical segment is th2(R-3h).

PROBLEMS IN MAXIMA AND MINIMA IN PLANE AND SOLID GEOMETRY

226. Through a given point draw a straight line which shall form with two given lines a triangle of minimum area.

227. Through a given point within a given angle draw a straight line which shall form with the sides of the given angle a triangle of minimum perimeter.

228. Through the intersection of two tangents to a circle draw a straight line cutting the circumference in two points such that, if they are joined to the points of tangency, the product of either pair of opposite sides of the inscribed quadrilateral thus formed shall be a maximum. 229. In an acute-angled triangle inscribe a rectangle, such that its diagonal shall be a minimum.

230. From a given point in a diameter AB of a circle produced draw a straight line cutting the circumference in two points C and H, so that the triangle ACH shall be a maximum.

231. Two straight lines containing a given angle are drawn from a given point in the base of a triangle, forming a quadrilateral with the two other sides. Prove that, of all the quadrilaterals which may be thus formed, that one whose sides passing through the given point are equal is a maximum, if the given angle is less than the supplement of the opposite angle of the triangle; a minimum, if the given angle is greater than the supplement of the opposite angle; neither a maximum nor a minimum, if the given angle is equal to the supplement of the opposite angle.

232. In the last exercise a maximum or a minimum quadrilateral can be formed for each point in the base (except in the case when the given angle is the supplement of the opposite angle of the triangle). Prove that, of all these maximum or minimum quadrilaterals, the least maximum or the greatest minimum is that whose equal sides make equal angles with the base.

233. Find a point in a plane such that the sum of its distances from two fixed points on the same side of the plane shall be a mini

mum.

234. Find a point in a plane such that the difference of its distances from two fixed points on opposite sides of the plane shall be a maximum.

235. Of all quadrangular prisms of which the volumes are equal, the cube has the least surface.

INTRODUCTION TO

MODERN GEOMETRY

[The numbers of the figures are the same as of the articles to which they belong.]

DIVISION OF LINES. THE COMPLETE QUADRILATERAL

1. The lines connecting any point with the three vertices of a triangle so divide the opposite sides that the product of three nonadjacent segments is equal to the product of the other three nonadjacent segments.

2. Conversely, if the sides of a triangle are so divided (either two or not any of the points of division being on the sides produced) that the product of three non-adjacent segments is equal to the product of the other three non-adjacent segments, the lines connecting the points of division with the opposite vertices meet in a point.

Hint.-Use the method of reductio ad absurdum.

3. Def.-A line which cuts a system of lines is a transversal.

In § 4 and § 5 XZ is a transversal which cuts the lines AB, AC, BC. 4. If the sides of a triangle are cut by a transversal, the product of three non-adjacent segments is equal to the product of the other three non-adjacent segments.

5. Conversely, if the sides of a triangle are so divided (either one or three of the points of division being on the sides produced) that the product of three non-adjacent segments is equal to the product of the other three non-adjacent segments, the points of division are in a straight line.

Hint. Use the method of reductio ad absurdum.

6. Exercise. If ABCD be four points taken in order on a straight line, AB.CD+BC.AD=AC.BD.

7. Def.-A complete quadrilateral is the figure formed by four straight lines intersecting in six points. The six points are the vertices; the three lines connecting opposite vertices are the diagonals. ABCDEF is a complete quadrilateral; AD, BE, CF, are the diagonals.

M

FIGS 7 AND 8

8. The middle points of the diagonals of a complete quadrilateral are in a straight line.

Hint.—Let L, M, N be the middle points of the diagonals. Construct the triangle X YZ, whose vertices are at the middle points of BD, DC, CB; the sides of this triangle pass through L, M, N.

Since FA is a transversal cutting the sides of the triangle BCD,

Therefore the points L, M, N, being on the sides of the triangle X YZ, are in a straight line.

ន 5

HARMONIC SECTION

9. Def.—If a line AB is divided harmonically at C and D, the points C and D are harmonic conjugates to the points A and B. The four points A, B, C, D are harmonic points, and AB is a harmonic mean between AC and AD.

A line is divided harmonically if it is divided internally and externally in the same ratio. § 332, p. 151

10. Exercise.-The above definition of a harmonic mean is equivalent to the algebraic definition.

Hint.-In Algebra the harmonic mean between a and b is

2ab

a + b