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be positive real numbers such that 
Find the minimum value of 
lies on the bisector of an acute angle with vertex 
. Let 
, 
be the projections of to the sidelines of the angle. The circle centered at with radius 
meets the sidelines at points 
and 
such that 
. Prove that the circle with center touching 
and the circle with center touching 
are tangent.
are nonzero real numbers such that![\[
\frac{1}{abc} + \frac{1}{a} + \frac{1}{c} = \frac{1}{b},
\]](http://latex.artofproblemsolving.com/6/d/9/6d9aabd8130b99d7128de9a3f063eb4c7124febf.png)
let 
be the maximum value of the expression![\[
\frac{4}{a^2 + 1} + \frac{4}{b^2 + 1} + \frac{7}{c^2 + 1}.
\]](http://latex.artofproblemsolving.com/a/b/2/ab28769126365574d4c0108500c370826082de6b.png)
Determine the sum of the numerator and denominator of the simplified fraction representing .
and 
are tangent to the circle 
at points and , respectively, and intersect at points and 
(
is not the diameter of ). The lines and 
are perpendicular. Prove that the radii of the circles and are the same.
and are tangent to the circle at points and , respectively, and intersect at points and ( is not the diameter of ). The lines and are perpendicular. Prove that the radii of the circles and are the same.
and 
Prove that
