# Proposition: 1.07: Uniqueness of Triangles

### (Proposition 7 from Book 1 of Euclid's “Elements”)

On the same straight line, two other straight lines equal, respectively, to two (given) straight lines (which meet) cannot be constructed (meeting) at a different point on the same side (of the straight line), but having the same ends as the given straight lines.

### Modern Formulation

Suppose two distinct triangles $$\triangle{ABC}$$ and $$\triangle{ABD}$$ share a common base $$\overline{AB}$$ and lie on the same side of it. Also suppose that at one endpoint of the base, e.g. $$A$$, the two sides connecting to $$A$$ are equal in length: $$\overline{AC}=\overline{AD}$$. Then the lengths of the sides connected to the other endpoint of the base (point $B$), are unequal in length: $$\overline{BC}\neq\overline{BD}$$.

Proofs: 1

Motivations: 1
Proofs: 2

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### References

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

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