Unique SSS Triangle

Theorem: Given the length of three sides of a triangle, a unique triangle can be formed.

prerequisites:
Isosceles triangle theorem (proof)

Proof:

Let $\triangle ABC$ be a triangle with the given side lengths. If possible, let $\triangle A'BC$ be another triangle such that A'B = AB and A'C = AC. Join the points A and A' by a straight line.

Now in $\triangle ABA'$,

$\qquad\quad AB = A'B$

Hence $\triangle ABA'$ is an isosceles triangle. Thus, by the isosceles triangle theorem,

$\qquad\quad\!\!\angle BAA' = \angle BA'A$
$\therefore\quad\;\;\alpha + \beta = \delta$
$\therefore\quad\;\;\delta > \beta\qquad\qquad\qquad\qquad\qquad\cdots\text{(1)}$

Similarly, in $\triangle ACA'$,

$\qquad\quad AC = A'C$

Hence $\triangle ACA'$ is an isosceles triangle. Thus, by the isosceles triangle theorem,

$\qquad\quad\!\!\angle CAA' = \angle CA'A$
$\therefore\quad\;\;\beta = \delta + \gamma$
$\therefore\quad\;\;\delta < \beta$

But this contradicts $(1)$.
Hence, it is not possible to construct a distinct triangle $\triangle A'BC$.

Thus, the triangle $\triangle ABC$ is unique.


Recommended:
Unique SAS triangle
Unique RHS triangle
SSS congruence