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Proposition 28

If a parallelepipedal solid is cut by a plane through the diagonals of the opposite planes, then the solid is bisected by the plane.
Let the parallelepipedal solid AB be cut by the plane CDEF through the diagonals CF and DE of opposite planes. I say that the solid AB is bisected by the plane CDEF. |
| Since the triangle CGF equals the triangle CFB, and ADE equals DEH, while the parallelogram CA equals the parallelogram EB, for they are opposite, and GE equals CH, therefore the prism contained by the two triangles CGF and ADE and the three parallelograms GE, AC, and CE equals the prism contained by the two triangles CFB and DEH and the three parallelograms CH, BE, and CE, for they are contained by planes equal both in multitude and in magnitude. | I.34 XI.Def.10
Hence the whole solid AB is bisected by the plane CDEF.
Therefore, if a parallelepipedal solid is cut by a plane through the diagonals of the opposite planes, then the solid is bisected by the plane.
Q. E. D.

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A minor point missing from the beginning of the proof of is that the two diagonals CF and DE lie in one plane, but it is easy to show that they lie in the lines CD and EF are parallel, and therefore, by XI.7, CF and DE lie in the plane spanned by CD and EF.

This is the second proposition concerning volumes. (The first was XI.25.) The final conclusion of the proof here is justified by XI.Def.10: since the faces of the two prisms are congruent, therefore the prisms are equal and similar (that is, congruent). Several authors have criticized this conclusion because the two prisms are mirror images of each other and cannot be applied to each other in the sense of moving one in space to coincide with the other.

From some points of view this criticism is valid. But the method of superposition is subject to even greater criticism. In modern geometry, depending on the style of geometry, superposition is either eliminated entirely or else completely formalized using the theory of group transformations.

#### Use of this proposition

Although this proposition is not used in the rest of this book, it is used for several propositions in the next book that deal with triangular prisms.

Book XI Introduction - Proposition XI.27 - Proposition XI.29.

© 1996
D.E.Joyce
Clark University