Proposition 37

Theorem

Two triangles (△ABC, △DBC) on the same base (BC) and between the same parallels (BC ‖ AD) have the same area.

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Commentary

Let △ABC and △DBC be two triangles that are on the same base (BC) and between the same parallel lines (BC and AD).

Then these two triangles are equal in area.

This proposition is a special case of the next proposition, Book 1 Proposition 38, and is the converse of Book 1 Proposition 39.

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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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