how can I prove that if $p$, $q$, $r$ and $o$ are points in a Hilbert space such that $p$, $q$, $o$ are collinear, $\|p-o\|=\|q-o\|$ and $\|p-r\|=\|q-r\|$ then $r-o \perp p-o$?.
I think it's a simple count but I can't do, can anybody help me?
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$$|p-r|^2 =| (p-o)+(o-r)|^2=|p-o|^2+|o-r|^2 + 2\langle p-o,o-r\rangle$$ Then write down the same for $|q-r|^2$, and compare the formulas. Also recall that $p-o= - (q-o)$. – Nov 26 '14 at 04:37