# Proposition: 1.37: Triangles of Equal Area I

### (Proposition 37 from Book 1 of Euclid's “Elements”)

Triangles which are on the same base and between the same parallels are equal to one another.

• Let $ABC$ and $DBC$ be triangles on the same base $BC$, and between the same parallels $AD$ and $BC$.
• I say that triangle $ABC$ is equal to triangle $DBC$.

### Modern Formulation

Triangles ($$\triangle{ABC}$$, $$\triangle{DBC}$$) on the same base ($$\overline{BC}$$) and standing between the same parallels ($$\overline{AD}$$, $$\overline{BC}$$) are equal in area.

Proofs: 1

Proofs: 1 2

Thank you to the contributors under CC BY-SA 4.0!

Github:

non-Github:
@Calahan
@Casey
@Fitzpatrick

### References

#### Adapted from CC BY-SA 3.0 Sources:

1. Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014

#### Adapted from (Public Domain)

1. Casey, John: "The First Six Books of the Elements of Euclid"

#### Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"