Proposition 35

Theorem

Two parallelograms (◇ABCD, ◇BCEF) on the same base (BC) and between the same parallels (BC ‖ AE) have the same area.

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Commentary

Let ◇ABCD and ◇BCEF be two parallelograms that are on the same base (BC) and between the same parallel lines (BC and AE).

Then these two parallelograms are equal in area.

This proposition is a special case of the next proposition, Book 1 Proposition 36.

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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

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