All triangles with equal base and equal area have equal height | Euclid's Elements Book 1 Prop 39

Claim: it is not possible for there to exist two triangles with equal area and equal bases but differing heights.Show more

Argument:

1. Let tri(A, B, C) and tri(D, B, C) be two triangles with equal area. (i.e. equal bases)
2. Our goal is to show the triangles have the same height.
3. Suppose join(A, D) is not parallel to join(B, C) (i.e. the triangles have equal area and equal bases but different heights). Our goal is to show this is impossible.
4. Construct E incident with join(B, D) such that join(A, E) is parallel to join(B, C).
5. Because tri(A, B, C) and tri(E, B, C) have the same base and the same height, they have the same area
6. However, either
6a. area(tri(E, B, C)) < area(tri(D, B, C)) [as shown in the diagram]
6b. area(tri(E, B, C)) > area(tri(D, B, C))
7. We have proven
7a. area(tri(A, B, C)) =:= area(tri(D, B, C)) [by assumption]
7b. area(tri(A, B, C)) =:= area(tri(E, B, C)) [by construction]
7b. area(tri(D, B, C)) =/= area(tri(E, B, C)) [by construction]
8. Therefore it is not possible for there to exist two triangles with equal area and equal bases but differing heights.

Element page: https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI39.html
GeoGebra: https://www.geogebra.org/m/d3fpquss

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