Problem
GEO-B2-M09-P007 The Median Formula
#7
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Let \(A(-1,0)\), \(B(1,0)\), \(C(u,v)\), and let \(M\) be the midpoint of \(AB\). Prove that \(CA^2+CB^2=2CM^2+2\).
Here \(M=(0,0)\). Expand the two squares.
\(CA^2=(u+1)^2+v^2\), \(CB^2=(u-1)^2+v^2\). Thus \(CA^2+CB^2=2u^2+2v^2+2=2CM^2+2\), since \(CM^2=u^2+v^2\).
This is a special form of Apollonius' theorem.