Vector Proof of Sylvester's Triangle Theorem pending

Prove that H = A + B + C in triangle ABC where Circumcenter O is the origin for the vector space and H,A,B,C are position vectors of the Orthocenter and vertices A,B,C respectively.

Proof:
For position vectors a,b,c:
|a| = |b| = |c| = R = circumradius
And
a.a = b.b = c.c = R^2

Let a vector h = a + b + c.
We will show that line segment from any vertex to H is perpendicular to the opposite side.
AH = OH - OA = h - a = a + b + c - a = b + c
BC = OC - OB = c - b

Compute dot product of AH and BC
AH.BC = (b + c) ( c - b) = b.c + c.c - b.b - c.b = c.c - b.b + b.c - c.b
Since b.c = c.b
AH.BC = c.c - b.b = R^2 - R^2

So AH is perpendicular to BC.
Similarly for other vertices and their opposite sides.
Since H lies on each altitude, it's the orthocenter.

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