Using the FLINE form for the equation for a straight line, I will show the the Three Point Aitken Constructions for points on a parabola is exactly equivalent to the three point Lagrange Polynomial.

The FLINE form provides a compact expression for the parametric equation for the straight line from the left-most point (XL,YL) to the right-most point (XR,YR).

Y=FLINE(XL,YL,XR,YR,X)=(YL(XR-X)+YR(X-XL))/(XR-XL)

Using this form the Aitken Construction for points on the parabola that passes through the points (X1,Y1), (X2,Y2) and (X3,Y3) is then

Y123=FLINE(X1,Y12,X3,Y23,X)

Y12=FLINE(X1,Y1,X2,Y2,X)

Y23=FLINE(X2,Y2,X3,Y3,X)

Equivalence_Parabola

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