G.530 solid contained parallel — Heath

XI.24

If a solid be contained by parallel planes, the opposite planes in it are equal and parallelogrammic.

For let the solid < C D H G > be contained by the parallel planes < A C >, < G F >, < A H >, < D F >, < B F >, < A E >; I say that the opposite planes in it are equal and parallelogrammic.

For, since the two parallel planes < B G >, < C E > are cut by the plane < A C >,

their common sections are parallel. [G.519]

Therefore / A B / is parallel to / D C /.

Again, since the two parallel planes < B F >, < A E > are cut by the plane < A C >,

[G.519]their common sections are parallel.

Therefore / B C / is parallel to / A D /.

But / A B / was also proved parallel to / D C /;

therefore < A C > is a parallelogram.

Similarly we can prove that each of the planes < D F >, < F G >, < G B >, < B F >, < A E > is a parallelogram.

Let / A H /, / D F / be joined.

Then, since / A B / is parallel to / D C /, and / B H / to / C F /, the two straight lines / A B /, / B H / which meet one another are parallel to the two straight lines / D C /, / C F / which meet one another, not in the same plane;

therefore they will contain equal angles; [G.512] therefore the angle < A B H ) is equal to the angle < D C F ).

And, since the two sides / A B /, / B H / are equal to the two sides / D C /, / C F /, [G.62] and the angle < A B H ) is equal to the angle < D C F ),

therefore the base / A H / is equal to the base / D F /, and the triangle < A B H > is equal to the triangle < D C F >. [G.24]

And the parallelogram < B G > is double of the triangle < A B H >, and the parallelogram < C E > double of the triangle < D C F >; [G.62]

therefore the parallelogram < B G > is equal to the parallelogram < C E >.

Similarly we can prove that < A C > is also equal to < G F >, and < A E > to < B F >.

Therefore etc. Q. E. D.

Dependency Graph

Required for

G.531, G.541, G.542, G.543, G.544, G.546, G.547, G.550, G.554, G.555, G.556, G.557, G.558, G.559, G.560, G.563, G.565, G.570, G.571, G.572, G.573, G.577, G.578