Proposition 36

Theorem

Two parallelograms (◇ABCD, ◇EFGH) on equal bases (ￜBCￜ = ￜFGￜ) and between the same parallels (BG ‖ AH) have the same area.

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Commentary

Let ◇ABCD and ◇EFGH be two parallelograms that are on equal bases (BC and FG) and between the same parallel lines (BG and AH).

Then these two parallelograms are equal in area.

This proposition is a generalization of the previous proposition, Book 1 Proposition 35.

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Computable version

In[1]:=

GeometricScene[{{A.,B.,C.,O.,D.,E.,F.,G.},{}},​​{GeometricStep[{CircleThrough[{A.,B.,C.},O.],Line[{A.,O.,C.}],Line[{D.,E.}],EuclideanDistance[D.,E.]≤EuclideanDistance[A.,C.]}],​​GeometricStep[{F.∈Line[{A.,C.}],EuclideanDistance[A.,F.]EuclideanDistance[D.,E.]}],​​GeometricStep[{GeometricAssertion[{CircleThrough[{F.},A.],CircleThrough[{A.,B.,C.},O.]},{"Concurrent",G.}]}],​​GeometricStep[{Line[{A.,G.}]}]},​​{EuclideanDistance[A.,G.]EuclideanDistance[D.,E.]}​​]//RandomInstance

Out[]=

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