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PROPOSITION VIII. THEOREM

451. Of two equivalent regular polygons, that which has the greater number of sides has the smaller perim

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452. COR. The circumference of a circle is less than the perimeter of any equivalent polygon.

MISCELLANEOUS EXERCISES

Ex. 1023. If two equal lines are divided externally so that the product of the segments of one is equal to the product of the segments of the other, the segments are equal respectively.

* Ex. 1024. Two triangles are equal if the base, the opposite angle, and its bisector of one are respectively equal to the base, the opposite angle, and its bisector of the other.

HINT. Circumscribe circles, produce bisectors until they meet the circumference, and join the point of intersection to one end of the base. (Ex. 733 and Ex. 1023.)

Ex. 1025. If two bisectors of a triangle are equal, the corresponding sides are equal, and the triangle is isosceles. (Ex. 1024.)

* Ex. 1026. The square of the side of a regular pentagon increased by the square of one of its diagonals is equal to five times the square of the radius.

* Ex. 1027. A regular pentagon is equivalent to a rectangle having one side equal to 5 times the radius, and the other to of a diagonal of the pentagon.

* Ex. 1028. The product of the diagonals of an inscribed quadrilateral is equal to the sum of the products of the opposite sides. (Ptolemy's Theorem.)

* Ex. 1029. To construct a triangle having given the three feet of the altitudes.

*Ex. 1030. If pn and P, are respectively the perimeters of an inscribed and circumscribed regular polygon of n sides, and p2n and P2, the perimeters of regular polygons of 2n sides, respectively inscribed and circumscribed about the same circle, prove that Pan is the harmonical mean between p, and P1; i.e.

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* Ex. 1031. Using the notations of Ex. 1030, prove that på, is the mean proportional between pn and P2n; i.e. P2n =` = √PnP2n

Ex. 1032. If circles be circumscribed about the four triangles into which a quadrilateral is divided by its diagonals, their centers form the vertices of a parallelogram.

SOLID GEOMETRY

BOOK VI

LINES AND PLANES IN SPACE-POLYEDRAL

ANGLES

Geometry of space or solid geometry treats of figures whose elements are not all in the same plane. (15)

453. DEF. A plane is a surface such that a straight line joining any two points in it lies entirely in the surface. (11) A plane is determined by given points or lines, if only one plane can be drawn through these points or lines.

PROPOSITION I. THEOREM

454. A plane is determined:

(1) By a straight line and a point without the line. (2) By three points not in the same straight line. (3) By two intersecting straight lines.

(4) By two parallel straight lines.

E

XC

B

F

(1) To prove that a plane is determined by a given straight line AB and a given point C.

Turn any plane EF passing through AB about AB as an axis until it contains C.

If the plane, so obtained, be turned in either direction, it would no longer contain C.

Hence, the plane is determined by AB and C.

(2) To prove that three given points A, B, and C determine a plane.

Draw AB. Then AB and C determine the plane. (Case 1.)

C

B

(3) To prove that two intersecting straight lines AB and AC determine a plane.

[Proof by the student.]

D

B

(4) To prove that two parallel lines AB and CD determine a plane.

The parallel lines AB and CD lie in the same plane by definition.

Since AB and the point C determine a plane, the two parallels determine a plane.

455. COR. The intersection of two planes is a straight line. For the intersection cannot contain three points not in a straight line, since only one plane can be passed through three such points.

456. DEF. The foot of the line intersecting a plane is the point of intersection.

457. DEF. A straight line is perpendicular to a plane if it is perpendicular to every line drawn through its foot in the plane.

458. DEF. A plane is perpendicular to a line if the line is perpendicular to the plane.

459. DEF. A straight line and a plane are parallel if they do not meet, however far they may be produced.

460. DEF. Two planes are parallel if they do not meet, however far they may be produced.

PROPOSITION II. THEOREM

461. If a straight line is perpendicular to each of two lines at their point of intersection, it is perpendicular to the plane of those lines.

A

M

B

E

N

Hyp. AB is 1 to BC and BD at B.

To prove AB is to the plane MN containing BC and BD. Proof. In plane MN draw any line BE through B.

Draw CD meeting BE in E and produce AB to F, so that BF = AB.

Draw AC, AE, AD, CF, EF, and DF.

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