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Point of intersection of the two lines

Source: IMO Longlist 1989, Problem 46

September 18, 2008
analytic geometryalgebra unsolvedalgebra

Problem Statement

Let S be the point of intersection of the two lines l_1 : 7x \minus{} 5y \plus{} 8 \equal{} 0 and l_2 : 3x \plus{} 4y \minus{} 13 \equal{} 0. Let P \equal{} (3, 7), Q \equal{} (11, 13), and let A A and B B be points on the line PQ PQ such that P P is between A A and Q, Q, and B B is between P P and Q, Q, and such that \frac{PA}{AQ} \equal{} \frac{PB}{BQ} \equal{} \frac{2}{3}. Without finding the coordinates of B B find the equations of the lines SA SA and SB. SB.
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