Trying to formulate a proof for this claim related to triangle inequality

(From outside the box geometry)

Claim: The longest side in a triangle must be opposite from the largest angle

Idea: Suppose there is some line L where L is the longest side for some triangle T

Then, since neither of the remaining two sides of T can have a length greater than that of line L, the remaining one vertex of T, V, should be located among the area colored green.

Clearly, the angle opposite to line L would be at its smallest when vertex V is located at the edge of the area colored green. Clearly, when vertex V is located at any point along the edge of the area colored green, the resulting triangle would either equilateral or bilateral, with the angle at vertex V always being either bigger than or equal to all other two angles.

Clearly, these pieces be enough to prove the original claim, if properly fitted and formulated together. However, making a formal proof is a process which I'm not very familiar with. Could someone perhaps help me with this?

#Geometry

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$