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PROPOSITION XIII. THEOREM. 545. The four diagonals of a parallelopiped bisect each other.

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Let AG, EC, BH, and FD, be the four diagonals of

the parallelopiped A G.
We are to prove these four diagonals bisect each other.

Through the opposite and Il edges A E and C G pass a plane intersecting the II bases in the Il lines A C and E G. The section ACG E is a o,

(having its opposite sides II); ... its diagonals AG and EC bisect each other in the point 0.

§ 138 In like manner a plane passed through the opposite and I edges F G and A D will form a D AFGD,

whose diagonals A G and F D will bisect each other in the point 0.

§ 138 Also, a plane passed through the opposite and Il edges E H and B C will form a O EBC H,

whose diagonals EC and B H will bisect each other in the point 0. .. the four diagonals bisect each other at the point 0.

Q. E. D. 546. COROLLARY. The diagonals of a rectangular parallelopiped are equal.

547. Scholium. The point 0, in which the four diagonals intersect, is called the centre of the parallelopiped ; and it is evident that any straight line drawn through the point ( and terminated by two opposite faces of the parallelopiped is bisected at that point. Hence O is the centre of symmetry.

ON PYRAMIDS.

548. DEF. A Pyramid is a polyhedron one of whose faces is a polygon, and whose other faces are triangles having a common vertex and the sides of the polygon for bases.

. 549. DEF. The Base of a pyramid is the face whose sides are the bases of the triangles having a common vertex.

550. DEF. The Lateral faces of a pyramid are all the faces except the base.

551. DEF. The Lateral surface of a pyramid is the sum of its lateral faces.

552. Def. The Lateral edges of a pyramid are the intersections of its lateral faces.

553. DEF. The Basal edges of a pyramid are the intersections of its base with its lateral faces.

554. DEF. The Vertex of a pyramid is the common vertex of its lateral faces.

555. DEF. The Altitude of a pyramid is the perpendicular distance from its vertex to the plane of its base.

Thus, V-A B C D E is a pyramid; A B C D E is its base ; A V B, BVC, etc. are its lateral faces, and their sumok is its lateral surface ; VA, V B, etc. V are its lateral edges ; A B, BC, etc. its basal edges ; V is its vertex ; VO is its altitude.

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556. DEF. A Regular pyramid is a pyramid whose base is a regular polygon, and whose vertex is in the perpendicular to the base at its centre.

557. DEF. The Axis of a regular pyramid is the straight line joining its vertex with the centre of the base.

558. DEF. The Slant height of a regular pyramid is the altitude of any lateral face.

559. DEF. A pyramid is triangular, quadrangular, pentangular, etc. according as its base is a triangle, quadrilateral, pentagon, etc. A triangular pyramid formed by four faces (all of which are triangles) is a tetrahedron.

560. DEF. A Truncated pyramid is the portion of a pyramid included between its base and a section cutting all its lateral edges.

561. DEF. A Frustum of a pyramid is a truncated pyramid in which the cutting section is parallel to the base.

562. DEF. The base of the pyramid is called the Lower base of the frustum, and the parallel section, its Upper base.

563. DEF. The Altitude of a frustum is the perpendicular distance between the planes of its bases.

564. DEF. The lateral faces of a frustum of a regular pyramid are the trapezoids included between its bases ; the lateral surface is the sum of the lateral faces; the Slant height of a frustum of a regular pyramid is the altitude of any lateral face.

PROPOSITION XIV. THEOREM. 565. If a pyramid be cut by a plane parallel to its base,

I. The edges and altitude are divided proportionally ; 11. The section is a polygon similar to the base.

Let the pyramid V-A B C D E, whose altitude is VO,

be cut by a plane abcde parallel to its base, in-
tersecting the lateral edges in the points a, b, c, d, e,
and the altitude in o.

We are to prove
Va V6

Vo
VA VB'.. = VOI
II. The section abcd e similar to the base A B C D E.
I. Suppose a plane passed through the vertex V ll to the base.

Then the edges and the altitude will be intersected by three
Il planes.
Va Vb

[ Ꮴo .
•. V A = VB..= vO'.

$ 469 (if straight lines be intersected by three Il planes, their corresponding segments

are proportional). II. The sides ab, bc etc. are parallel respectively to A B, BC,

§ 465 (the intersections of II planes by a third plane are Il lines) ; .. Is abc, bcd etc. are equal respectively to A BC, B C D etc.,

$ 462 (if two is not in the same plane have their sides respectively Il and lying in

the same direction, they are equal).
.. the two polygons are mutually equiangular.

etc.,

Also, since the sides of the section are Il to the corresponding sides of the base,

Vab, Vbc etc. are similar respectively to A VAB, VB-C etc.

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.. the polygons have their homologous sides proportional;
.. section abcde is similar to the base A B C D E 278

Q. E. D. 566. COROLLARY 1. Any section of a pyramid, parallel to its base is to the base as the square of its distance from the vertex is to the square of the altitude of the pyramid.

VO (V6 ab
Since

yo =yB FAB

V 2 a 12
Squaring

V U2Ā B2

a b c d e a 12
But
A B C D E Ā 32,

§ 344 (similar polygons are to each other as the squares of their homologous sides).

abcde V2

.. A B C D E VO2 • 567. Cor. 2. If two pyramids having equal altitudes be cut by planes parallel to their bases, and at equal distances from their vertices, the sections will have the same ratio as their bases.

a b c d e V 2 For

ABC DEV02,

ale vor

A'B'C V1012 . Now, since Vo=V'd', and V (= O',

abcde : ABCDE ::a'b' ¢ : A' B'C'. Whence abcde : a' b'¢ : : A B C D E : A' B'C'. $ 262

568. COR. 3. If two pyramids have equal altitudes and equivalent bases, sections made by planes parallel to their bases and at equal distances from their vertices are equivalent.

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