# Length bisection from circular arc

I am not sure if the following result is well known. I stumbled across it from the paper The Perimetric Bisection of Triangles by Dov Avishalom, where the result was stated without proof. I am looking for a simple proof of the following

The following figure depicts a circular arc with chord $\mathrm{LN}$. The point $\mathrm{P}$ denotes the midpoint of the circular arc. We drop perpendicular from $\mathrm{P}$ to $\mathrm{LM}$, intersecting it at point $\mathrm{Q}$.

Then the claim is that $\mathrm{Q}$ bisects the broken line segment $\mathrm{LMN}$, that is we have $\mathrm{LQ} = \mathrm{QM}+\mathrm{MN}$.

Thanks for any help.

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