AoPS - Problem

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Problem

Source: Czech-Polish-Slovak Junior Match 2015, Team p3 CPSJ

Tags: geometry, geometric inequality, equal segments, perpendicular

parmenides51

19.03.2020 17:01

Different points $A$ and $D$ are on the same side of the line $BC$ , with $|AB| = | BC|= |CD|$ and lines $AD$ and $BC$ are perpendicular. Let $E$ be the intersection point of lines $AD$ and $BC$ . Prove that $||BE| - |CE|| < |AD| \sqrt3$

kiyoras_2001

04.07.2020 12:39

Let $BC=1, EC=x<1/2$ . Then $BE=1-x, AE=\sqrt{1-(1-x)^2}, DE=\sqrt{1-x^2}$ and we need to show that $1-2x<\sqrt{3}(\sqrt{1-x^2}-\sqrt{1-(1-x)^2})$ $\iff$ $1-2x<\sqrt{3}\cdot \frac{1-2x}{\sqrt{1-x^2}+\sqrt{1-(1-x)^2}}$ $\iff$ $\sqrt{1-x^2}+\sqrt{1-(1-x)^2}<\sqrt{3}$ $\iff$ $\sqrt{(1-x^2)(1-(1-x)^2)}<1+x$ $\iff$ $0<(1-x)^3+2x.$