NOTATION:
PQ denotes the vector from point P to point Q.
⊗ denotes the vector cross product.
CLAIM:
Let E and F be the midpoints of diagonals AC and
BD respectively. Since ABCD is not a parallelogram,
points E and F are distinct. The locus of points P
is the line EF.
PROOF:
[APB] + [CPD] = [BPC] + [DPA]
<==> PA ⊗ PB + PC ⊗ PD = PB ⊗ PC + PD ⊗ PA
<==> PA ⊗ PB + PA ⊗ PD + PC ⊗ PB + PC ⊗ PD = 0
<==> PA ⊗ (PB + PD) + PC ⊗ (PB + PD) = 0
<==> (PA + PC) ⊗ (PB + PD) = 0
<==> (PE + EA + PE + EC) ⊗ (PF + FB + PF + FD) = 0
<==> PE ⊗ PF = 0
<==> P lies on line EF.
QED
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