Triangles on the Same Base Between Parallels Are Equal
A Euclidean proof: two triangles, each half of an equal parallelogram
Following Euclid, Leonardo demonstrates that all triangles constructed upon the same base and between the same parallel lines are equal in area. He takes triangles a b c and d b c on base b c between the parallels a h and b f, completes them into parallelograms by drawing e c and g c, and argues that since each triangle is half of an equal parallelogram, the two triangles must themselves be equal. A diagram at the head of the page lays out the base line with the paired triangles between the two parallels.
On this page
Equal triangles a b c and d b c on base b c between parallels a h and b f
Leonardo asserts that all triangles built on the same base and between parallel lines are equal. To prove it for triangles a b c and d b c he draws e c parallel to b d and g c parallel to a b, forming two parallelograms shown equal by the 36th proposition. Because the diagonals a c and d c bisect these equal parallelograms, the two triangles, each a half, are equal to one another.