Problem
GEO-B2-M09-P004 Diagonals of a Parallelogram
#4
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Let \(A(0,0)\), \(B(u,v)\), \(D(p,q)\), \(C(u+p,v+q)\). Prove that diagonals \(AC\) and \(BD\) are bisected by the same point.
Find the midpoints of \(AC\) and \(BD\).
The midpoint of \(AC\) is \(\left(\frac{u+p}{2},\frac{v+q}{2}\right)\). The midpoint of \(BD\) is also \(\left(\frac{u+p}{2},\frac{v+q}{2}\right)\). They coincide, so the diagonals bisect each other.
A coordinate form of the parallelogram criterion.