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Limit of radius of inscribed circle

Consider the points A(0, 0) , B(1, 0) , C(x, 1)  with x>0. Let r(x) be the radius of the inscribed circle of the triangle ABC.

Prove that

    \[\lim_{x \rightarrow +\infty} \rho(x)=0\]

Solution

Since \tan A = \frac{1}{x} we deduce that \hat{A} \rightarrow 0 as x \rightarrow +\infty. Since the incenter I lies on the bisector of A , it follows that if D is the projection of I on the x'x axis

\rho = AD \tan \frac {A}{2}\leq 1\tan \frac {A}{2} \rightarrow 0

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