Really interesting and tough kinematics problem

TSny

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I'm pretty sure that @PeroK 's origin is at the center of the triangle.
OK, thanks. I had noticed in post #16 that @PeroK might be taking the center of the triangle as the origin. But, if so, then $|OA| = |OB| = |OC|$ and the following statement would truly be obvious.
... you can show the stronger condition that:

$|OA|^2 \le |OB|^2 + |OC|^2$

This looks fairly obvious and can be shown with a little algebra.

haruspex

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I'm pretty sure that @PeroK 's origin is at the center of the triangle.
No, it is not. See post #22.

Fun fact: equality is achieved if the origin is anywhere on the circle through the vertices of the triangle.

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PeroK

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Is there a restriction on the location of the origin? If not, then I don't think this relation is always true. For example, take the origin O to be the midpoint between B and C.

Anyway, I'm still trying to work out your approach. So far, no luck. But I will keep thinking about it. Fun problem.
You're right. I made a mistake in simplifying the expressions. It's still tricky!