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x_1 + x_4 =x_2+x_3, y_1y_4 =y_2y_3 wanted, points on y = mx-k^2

Source: 2010 Swedish Mathematical Competition p2

April 30, 2021
analytic geometry

Problem Statement

Consider the four lines y=mx−k2y = mx-k^2 for different integer kk. Let (xi,yi)(x_i,y_i), i=1,2,3,4i = 1,2,3,4 be four different points , such that each belongs to two different lines and on each line pass through just the two of them. Lat x1≤x2≤x3≤x4x_1\leq x_2\leq x_3\leq x_4. Show that x1+x4=x2+x3x_1 + x_4 =x_2+x_3 and y1y4=y2y3y_1y_4 =y_2y_3.
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