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To be the projection, or the be projected, that's the q!

Source: Flanders MO 2006 Q2

April 19, 2006
geometrygeometric transformationreflectiontrigonometryalgebralinear equation

Problem Statement

Let △ABC\triangle ABC be an equilateral triangle and let PP be a point on [AB]\left[AB\right]. QQ is the point on BCBC such that PQPQ is perpendicular to ABAB. RR is the point on ACAC such that QRQR is perpendicular to BCBC. And SS is the point on ABAB such that RSRS is perpendicular to ACAC. Q′Q' is the point on BCBC such that PQ′PQ' is perpendicular to BCBC. R′R' is the point on ACAC such that Q′R′Q'R' is perpendicular to ACAC. And S′S' is the point on ABAB such that R′S′R'S' is perpendicular to ABAB. Determine ∣PB∣∣AB∣\frac{|PB|}{|AB|} if S=S′S=S'.
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