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x_1/y_1 + x_2/y_2 + x_3/y_3 = 0 if 3 collinear point lie on y^2 = x^3

Source: 1992 Swedish Mathematical Competition p6

April 2, 2021
analytic geometrycollinear

Problem Statement

(x1,y1),(x2,y2),(x3,y3)(x_1, y_1), (x_2, y_2), (x_3, y_3) lie on a straight line and on the curve y2=x3y^2 = x^3. Show that x1y1+x2y2+x3y3=0\frac{x_1}{y_1} + \frac{x_2}{y_2}+\frac{x_3}{y_3} = 0.
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