High School Geometry - If $BC$ is the greatest side of $\triangle ABC$, $D$ & $E$ are points on $BC, CA$...

We assume as known that in any triangle $PQR$, the largest side is opposite the largest angle, and conversely.

Lemma: Let $PQR$ be a triangle, and let $X$ be in the interval $PR$, and not at an endpoint. Then $QX \lt QP$ or $QX \lt QR$ (or both). More compactly, $QX\lt \max(QP,QR)$.

Proof: Since angles $QXP$ and $QXR$ add up to $180^\circ$, one at least is $\ge 90^\circ$. So $\angle QXP$ is the largest angle in $\triangle QXP$, or $\angle QXR$ is the largest angle in $\triangle QXR$, or both. If $\angle QXP$ is the largest angle in $\triangle QXP$, then $QP \gt QX$. Similarly, if $\angle QXR$ is the largest angle in $\triangle QXR$, then $QR\gt QX$.

Now we look at our triangle $ABC$. Suppose that $D=B$. If $E=C$, then $DE=BE=BC$. If $E\ne C$, then by the Lemma $BE$ (that is, $DE$) is $\lt\max(BC,BA)=BC$.

Suppose now that $D\ne B$. The case $E=C$ is easy.

If $B\ne D$ and $E\ne C$, draw the line $BE$. By the Lemma, we have $BE\lt BC$. Again by the Lemma, $DE\lt \max(BE,CE)\lt BC$. This completes the proof.