distance between any two interior points in a triangle less than largest side

Question

How can one prove that the distance between any two interior points in a triangle less than largest side by elementary way using euclidian geometry.

PS : I'm aware of this answer which cannot be delivered to school students

Distance between interior points of triangle

Answer 1

How about something like this...

Lemma 1: If AB is the largest side of $\Delta ABC$, there is no interior point $M$ of $\Delta ABC$ such that $AM > AB$.

Proof: Suppose there exists such point $M$.
Draw circle with center $A$ and radius $AB$.

Since $AM > AB$ this means $M$ lies outside of the cirlce.

But on the other hand $M$ lies inside $\Delta ABC$ and hence in the circle. This is a contradiction.

Lemma 2: For any interior point $M$ of any triangle $ABC$, we have that $CM$ is smaller than $r = \max(CA,CB)$

Proof: Draw circle with center $C$ and radius $r$. Again: $M$ has to be inside $ABC$ but outside the circle - again contradiction.

Now let $K,L$ be any two points inside the triangle $ABC$.

Case 1) $L$ is inside $AKC$. Then $KL$ is smaller than the max of $KA$ and $KC$ (by lemma 2).
But $KA$ is smaller than the max of $AC$ and $AB$ (again by lemma 2).
And $KC$ is smaller than the max of $CA$ and $CB$ (again by lemma 2).

Case 2) $L$ is inside $BKC$. The argument here goes identical to case 1)

Case 3) $L$ is inside $AKB$
Then $KL$ is smaller than the max of $KA$ and $KB$. Then we apply lemma 2 to $KA$ and $KB$ (and the big triangle $ABC$) and we get the desired result.

Of course one needs to polish some edge cases e.g. the case when $L$ lies on the boundary of some of the smaller triangles.
But this proof as a whole should work.

I just realized I do not use Lemma 1 anywhere in the proof.

Answer 2

A pictorial proof:

The base B is the longest side of the triangle. If there exists a segment L bigger than it(B) that lies inside the triangle, we can join the endpoints to the vertices of the longest side (B) as shown. Also, we must have endpoint X of L inside the triangle. But, by basic proportionality theorem, the green segment(which is at most L) is smaller (a fraction) than the longest side B, so point X must lie outside the triangle always; so L can't lie inside the triangle.