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Putnam 1940 A8

Source: Putnam 1940

February 22, 2022
Putnamgeometrylinear algebramatrix

Problem Statement

A triangle is bounded by the lines a1x+b1y+c1=0a_1 x+ b_1 y +c_1=0, a2x+b2y+c2=0a_2 x+ b_2 y +c_2=0 and a2x+b2y+c2=0a_2 x+ b_2 y +c_2=0. Show that its area, disregarding sign, is Δ22(a2b3−a3b2)(a3b1−a1b3)(a1b2−a2b1),\frac{\Delta^{2}}{2(a_2 b_3- a_3 b_2)(a_3 b_1- a_1 b_3)(a_1 b_2- a_2 b_1)}, where Δ\Delta is the discriminant of the matrix M=(a1b1c1a2b2c2a3b3c3).M=\begin{pmatrix} a_1 & b_1 &c_1\\ a_2 & b_2 &c_2\\ a_3 & b_3 &c_3 \end{pmatrix}.
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