Mary McGregor Paolo Prendushi Proposition 41 Mary McGregor Paolo Prendushi

Proposition 41 If a parallelogram has the same base with a triangle and is in the same parallels, then the parallelogram is double the triangle.

Proof Let the parallelogram ABCD have the same base BC with the triangle EBC, and let it be in the same parallels BC and EC Join AC (Postulate 1) Then, the triangle ABC equals the triangle EBC (Prop 37)

Proof But the parallelogram ABCD is double the triangle ABC, for the diameter AC bisects it, so that the parallelogram ABCD is also double the triangle EBC. (Prop 34) Therefore if a parallelogram has the same base with a triangle and is in the same parallels, then the parallelogram is double the triangle.

Generally, if a parallelogram has an equal base with a triangle and is in the same parallels, then the parallelogram is double the triangle.