G.540 parallelepipedal solids base

XI.30

Parallelepipedal solids which are on the same base and of the same height, and in which the extremities of the sides which stand up are not on the same segments, are equal.

parallelepipedal_solids_base_xi_30(< C M >, < C N >, < A B >, / A F /)

given < C M >, < C N > be parallelepipedal solids on the same base < A B > and of the same height, let the extremities of their sides which stand up, namely / A F /, / A G /, / L M /, / L N /, / C D /, / C E /, / B H /, / B K /, not be on the same segments

prove the solid < C M > is equal to the solid < C N >

given < C M >, < C N > be parallelepipedal solids on the same base < A B > and of the same height, let the extremities of their sides which stand up, namely / A F /, / A G /, / L M /, / L N /, / C D /, / C E /, / B H /, / B K /, not be on the same segments

construct extend(/ N K /, / D H / be produced and meet one another at R, and further let / F M /, / G E /) → P, Q

construct segment(A, O) → / A O /, segment(L, P) → / L P /, segment(C, Q) → / C Q /, segment(B, R) → / B R /

assert Then the solid < C M >, of which the parallelogram < A C B L > is the base, and < F D H M > its opposite, is equal to the solid < C P >, of which the parallelogram < A C B L > is the base, and < O Q R P > its opposite

∵ parallelepipedal_solids_baseassert For they are on the same base < A C B L > and of the same height, and the extremities of their sides which stand up, namely / A F /, / A O /, / L M /, / L P /, / C D /, / C Q /, / B H /, / B R /, are on the same segments / F P /, / D R /

assert The solid < C P >, of which the parallelogram < A C B L > is the base, and < O Q R P > its opposite, is equal to the solid < C N >, of which the parallelogram < A C B L > is the base and < G E K N > its opposite

assert For they are again on the same base < A C B L > and of the same height, and the extremities of their sides which stand up, namely / A G /, / A O /, / C E /, / C Q /, / L N /, / L P /, / B K /, / B R /, are on the same segments / G Q /, / N R /

∴ the solid < C M > is also equal to the solid < C N >

Dependency Graph

Required for

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