If a parallelogram has the same base as a triangle and is in the same parallels, then the parallelogram is double the triangle.
[Desmos graphs can be found here]
We begin with parallelogram ABCD and triangle EBC sharing the same base BC and be in the same parallels BC and AE.
We will prove that the parallelogram ABCD is double the triangle EBC.
Start by joining AC.
Compare triangles ABC and EBC. They have the same base BC, and they are in the same parallels BC and AE, so the two triangles are equal by Proposition 37. (Triangles with the same base that are in the same parallels are equal to one another.)
Since AC is the diameter of the parallelogram ABCD, the parallelogram is double the triangle ABC by Proposition 34. (The diameter of a parallelogram bisects it.)
Since the parallelogram ABCD is double the triangle ABC, and the triangle ABC is equal to the triangle EBC, therefore the parallelogram ABCD is double the triangle EBC.
Q.E.D.
In the next proposition we return to a construction of a parallelogram that is equal to a given triangle – George