G.541 parallelepipedal solids equal

XI.31

Parallelepipedal solids which are on equal bases and of the same height are equal.

parallelepipedal_solids_equal(< A E >, < C F >, < A B >, < C D >)

given the parallelepipedal solids < A E >, < C F >, of the same height, be on equal bases < A B >, < C D >

prove the solid < A E > is equal to the solid < C F >

given the parallelepipedal solids < A E >, < C F >, of the same height, be on equal bases < A B >, < C D >

assert First, let the sides which stand up, / H K /, / B E /, / A G /, / L M /, / P Q /, / D F /, / C O /, / R S /, be at right angles to the bases < A B >, < C D >

construct extend(/ C R /) → / R T /

construct copy_angle: On / R T /, and at R on it, let < T R U ) be constructed equal to < A L B ), let / R T / be made equal to / A L /, and / R U / equal to / L B /, and let the base < R W > and the solid < X U > be completed.

assert / T R /, / R U / = / A L /, / L B /, and they contain equal angles,

assert the parallelogram < R W > is equal and similar to the parallelogram < H L >

assert again / A L / = / R T /, and / L M / to / R S /, and they contain right angles,

assert the parallelogram < R X > is equal and similar to the parallelogram < A M >

assert For the same reason < L E > is also equal and similar to < S U >

assert three parallelograms of the solid < A E > are equal and similar to three parallelograms of the solid < X U >

∵ solid_contained_parallelassert The former three are equal and similar to the three opposite, and the latter three to the three opposite

∵ equal_similar_solid_defassert the whole parallelepipedal solid < A E > is equal to the whole parallelepipedal solid < X U >

construct Let / D R /, / W U / be drawn through and meet one another at Y

construct parallel_line(T, / D Y /) → / a T b /

construct extend(/ P D /) → a

assert Let the solids < Y X >, < R I > be completed

∵ parallelepipedal_solids_baseassert Then the solid < X Y >, of which the parallelogram < R X > is the base and < Y c > its opposite, is equal to the solid < X U > of which the parallelogram < R X > is the base and < U V > its opposite, for they are on the same base < R X > and of the same height, and the extremities of their sides which stand up, namely / R Y /, / R U /, / T b /, / T W /, / S e /, / S d /, / X c /, / X V /, are on the same segments / Y W /, / e V /

assert The solid < X U > is equal to < A E > :

assert the solid < X Y > is also equal to the solid < A E >

∵ parallelograms_same_baseassert since the parallelogram < R U W T > is equal to the parallelogram < Y T > for they are on the same base / R T / and in the same parallels / R T /, / Y W /, while < R U W T > is equal to < C D >, since it is also equal to < A B >

assert the parallelogram < Y T > is also equal to < C D >

assert < D T > is another parallelogram

∵ equal_magnitudes_ratioassert as < C D > is to < D T >, so is < Y T > to < D T >

assert since the parallelepipedal solid < C I > has been cut by the plane < R F > which is parallel to opposite planes,

∵ parallelepipedal_solid_cutassert As < C D > is to < D T >, so is the solid < C F > to the solid < R I >

assert For the same reason, since the parallelepipedal solid < Y I > has been cut by the plane < R X > which is parallel to opposite planes,

∵ parallelepipedal_solid_cutassert As < Y T > is to < T D >, so is the solid < Y X > to the solid < R I >

assert But, as < C D > is to < D T >, so is < Y T > to < D T >

∵ ratios_ratioassert also, as the solid < C F > is to the solid < R I >, so is the solid < Y X > to < R I >

∵ magnitudes_ratio_equalassert each of the solids < C F >, < Y X > has to < R I > the same ratio; therefore the solid < C F > is equal to the solid < Y X >

assert < Y X > was proved equal to < A E >

assert < A E > is also equal to < C F >

assert Next, let the sides standing up, / A G /, / H K /, / B E /, / L M /, / C N /, / P Q /, / D F /, / R S /, not be at right angles to the bases < A B >, < C D >

assert I say again that the solid < A E > is equal to the solid < C F >

construct From K, E, G, M, Q, F, N, S let / K O /, / E T /, / G U /, / M V /, / Q W /, / F X /, / N Y /, / S I / be drawn perpendicular to the plane of reference

construct segment() → O, segment() → T, segment() → U, segment() → V, segment() → W, segment() → X, segment() → Y, segment() → I, segment(O, T) → / O T /, segment(O, U) → / O U /, segment(U, V) → / U V /, segment(T, V) → / T V /, segment(W, X) → / W X /, segment(W, Y) → / W Y /, segment(Y, I) → / Y I /, segment(I, X) → / I X /

∵ First part of this Prop.assert Then the solid < K V > is equal to the solid < Q I >, for they are on the equal bases < K M >, < Q S > and of the same height, and their sides which stand up are at right angles to their bases

∵ parallelepipedal_solids_base_xi_30assert The solid < K V > is equal to the solid < A E >, and < Q I > to < C F >; for they are on the same base and of the same height, while the extremities of their sides which stand up are not on the same segments

∴ the solid < A E > is also equal to the solid < C F >

Dependency Graph

Required for

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