## Euclid, Elements of Geometry, Book I, Proposition 41 (Edited by Dionysius Lardner, 1855)

Proposition XLI. Theorem.
[Euclid, ed. Lardner, 1855, on Google Books]

 (190) If a parallelogram (B D) and a triangle (B E C) have the same base and be between the same parallels, the parallelogram is double of the triangle.

Draw C A. The triangle B E C is equal to the triangle B A C (XXXVII); but B D is double of the triangle B A C (XXXIV), therefore B D is also double of the triangle B E C.

(191)   This proposition may be generalized thus: If a parallelogram and triangle have equal bases and altitudes, the parallelogram is double the triangle (175).

(192)   Also, If a parallelogram and a triangle have equal altitudes, and the base of the triangle be double the base of the parallelogram, the parallelogram and triangle will be equal (178).

(193)   If a parallelogram and triangle have equal bases, and the altitude of the triangle be double the altitude of the parallelogram, they will be equal.

Next: Proposition 42

Previous: Proposition 40

This proposition in other editions: