Moreover,

Substituting for V', ', B', b', their equals in (1), we get

V=h(B+b+√Bxb).

Q. E. D.

SIMILAR POLYEDRONS

14. Def-Two polyedrons are similar if they have the same number of faces similar each to each and similarly placed, and their homologous diedral angles are equal.

715. The ratio of any two homologous edges of two simi lar polyedrons is equal to the ratio of any other two homolo gous edges.

H

H'

GIVEN the similar polyedrons AH and A'H' in which any two edges AB and CH of one are respectively homologous to A'B' and C'H' of the other.

Since the faces ABGF and A'B'G'F' are similar,

AB BG

A'B' B'G'

Since the faces BCHG and B'C'H'G' are similar,

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Ax. I

Q. E. D.

716. Def.-The ratio of any two homologous edges of two similar polyedrons is called the ratio of similitude of the polyedrons.

717. COR. I. The ratio of any two homologous faces of two similar polyedrons is equal to the square of their ratio of similitude.

Hint.-Apply § 401.

718. COR. II. The ratio of the total surfaces of two similar polyedrons is equal to the square of their ratio of similitude. Hint.-Apply § 265.

719. Two polyedrons similar to a third are similar to each

GIVEN the polyedrons AH, or X, and A'H', or Y, both similar to PW, or Z.

TO PROVE

that X is similar to Y.

The faces AC and A'C', being both similar to PR, are similar to each other.

$294

In the same way all the faces of X may be shown similar to corresponding faces of Y.

The polyedrons X and Y also have the similar faces similarly arranged, since the arrangement in each is the same as the arrangement in Z.

Lastly, any two homologous diedral angles of X and Y, being each equal to the same diedral angle of Z, are equal to each other.

Therefore the polyedrons X and Y are similar.

Ax. I

$714

Q. E. D

720. Two similar polyedrons are equal, if their ratio of

The homologous faces are equal, being similar and having unity as a ratio of similitude.

Superpose the faces ABHK and A'B'H'K'.

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Since the diedral angles AK and A'K' are equal, the planes of the faces CAKG and C'A'K'G' will coincide, and since the side AK of one already coincides with the side A'K' of the other, these faces, being equal, will coincide through

out.

In this way all the faces can be shown to coincide.
Therefore the polyedrons are equal.

Q. E. D.

721. If two diedral angles have their faces respectively parallel and extending in the same direction, they are equal.

The proof is left to the student.

722. Defs. If the vertices A, B, C, D, etc., of a polyedron are joined by straight lines to any point O, and the lines OA, OB, OC, OD, etc., are divided in the same ratio at the points A', B', C', D', etc., the polyedron A'B'C'D', etc., is said to be radially situated with regard to the polyedron ABCD, etc.

The ratio of the rays OA' and OA is called the determining ratio, or ray ratio, of the two polyedrons.

The point

is called the ray centre.

PROPOSITION XXXI.

THEOREM

723. Two radially situated polyedrons are similar, and their ratio of similitude is equal to the ray ratio.