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d(A,ll)> d(P, AQ), distances inequality with 3 collinear points Chile 2010 L2 P5

Source:

May 31, 2019
geometrygeometric inequalitydistance

Problem Statement

Consider a line ℓ \ell in the plane and let B1,B2,B3 B_1, B_2, B_3 be different points in ℓ \ell. Let A A be a point that is not in ℓ \ell. Show that there is P,Q P, Q in B1,B2,B3 {B_1, B_2, B_3} with P≠Q P \ne Q so that the distance from A A to ℓ \ell is greater than the distance from P P to the line that passes through A A and Q Q .
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