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1/(a^2+1/2 )< b/a < 1/a^2 , geo inequality related to a circle

Source: 1997 Swedish Mathematical Competition p1

April 2, 2021

Geometric Inequalitiesgeometryinequalities

Problem Statement

Let

AC

be a diameter of a circle and

AB

be tangent to the circle. The segment

BC

intersects the circle again at

D

. Show that if

AC = 1

AB = a

, and

CD = b

, then

\frac{1}{a^2+ \frac12 }< \frac{b}{a}< \frac{1}{a^2}