Proposition 41

Theorem

If a parallelogram (◇ABCD) and a triangle (△EBC) are on the same base (BC) and between the same parallels, the area of the parallelogram is twice the area of the triangle.

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Commentary

Let ◇ABCD be a given parallelogram and △EBC be a given triangle. Let them both have the same base BC and be between the same parallel lines BC and AE.

Then the area of ◇ABCD is twice the area of △EBC.

This proposition generalizes one of the claims in Book 1 Proposition 34.

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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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