Since the triangles CAD and CA'E are isosceles by construction, and AF and A'G perpendiculars upon their bases,

CFCD, and CGCE.

But as CD is greater than CE, CF> CG.

CF is the altitude of the triangle BAC, and CG of BA'C, as these triangles have the same base, the one which has the greater altitude is the greater, or

area ABC > A'BC.

Q.E.D.

315. COR. Of all the triangles of the same perimeter, that which is equilateral is the maximum.

For the maximum triangle having a given perimeter must be isosceles whichever side is taken as the base.

PROPOSITION XV. THEOREM.

316. Of isoperimetric polygons having the same number of sides, that which is equilateral is the maximum.

A

B B'

E

Let ABCDE be an equilateral polygon.

To prove that it is greater than any other isoperimetric polygon.

If not greater, suppose AB'CDE is greater.

Draw AC.

Then ABC being an isosceles triangle, we know (by 314) area ABC > AB'C.

or

Add area ACDE to this inequality,

ACDE+ ABC > ACDE + AB'C,

ABCDE>AB'CDE.

That is, AB and BC cannot be unequal, and in like manner it can be shown that BC

equilateral.

=

CD DE, etc., or the polygon is

=

PROPOSITION XVI. THEOREM.

Q.E.D.

317. Of two isoperimetric regular polygons, that which has the greater number of sides has the greater area.

Let M be an equilateral triangle, and N an isoperimetric

Let D be any point in the side AC of the triangle.

Then the triangle M may be regarded as an irregular quadrilateral, having the four sides AB, BC, CD, and DA; the angle at D being equal to two right angles.

Hence, since the two quadrilaterals are isoperimetric,

In like manner, it may be proved that the area of a regular pentagon is greater than that of an isoperimetric square; that the area of a regular hexagon is greater than that of an isoperimetric regular pentagon; and so on.

Q.E.D.

318. COR.

Since a circle may be regarded as a regular polygon of an infinite number of sides, it follows that the circle is the maximum of all isoperimetric plane figures.

EXERCISES.

1. Of all triangles of given base and area, the isosceles is that which has the greatest vertical angle.

2. The shortest chord which can be drawn through a given point within a circle is the perpendicular to the diameter which passes through that point.

PROPOSITION XVII. THEOREM.

319. The sum of the distances from two fixed points on the same side of a straight line to the same point in that line is a minimum when the lines joining the fixed points with the same point are equally inclined to the given line.

Let CD be the straight line, A and B the fixed points, P such a point in CD that APC = ≤ BPD, and Q any other point in CD.

Let fall the perpendicular AF, and continue it until it meet BP produced, say in E, and join QA and QE. Since BE and CD are intersecting lines (by 49),

By hypothesis,

therefore

BPD FPE.

=

ZAPFBPD;

ZAPF = FPE.

The triangles APF and FPE are right triangles by construction, and having the side FP common, are (by 90) equal in all their parts; that is

NOTE. If CD is a reflecting surface, a ray of light in order to go from A to B by reflection, pursues the shortest path when the angle of incidence (APF) is equal to the angle of reflection (BPD). This is the physical law, thus furnishing one illustration of the economy in nature.

EXERCISES.

1. Given the base and the vertical angle of a triangle; to construct it so that its area may be a maxi

mui.

SUGGESTION. See 195, Ex. 1.

2. Show that the greatest rectangle which can be inscribed in a circle is a square.

SOLID GEOMETRY.

BOOK VI.

PLANES AND SOLID ANGLES.

DEFINITIONS.

320. A plane is (by 9) a surface such that a straight line which joins any two of its points will lie wholly in the surface.

321. A plane is of unlimited extent in its length and breadth; but to represent a plane in a diagram it is necessary to take only a definite portion, and usually it is represented by a parallelogram which is supposed to lie in the plane.

322. A plane is said to be determined by any combination of lines or points when it is the only plane which contains these lines or points.

323. Any number of planes may be passed through any given straight line.

For if a plane is passed through any given straight line AB, the plane may be turned about AB as an axis, and made to occupy an infinite number of positions, each of which will be a different plane passing through AB.

A

B

From this it can be seen that a single straight line does not determine a plane.