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Inscribed Circle In Right Angled Triangle

Problem

A circle is inscribed in a right angled triangle with the given dimensions.

Find the radius of the circle in terms of $a$, $b$, and $c$.


Solution

Consider the following diagram.

If we let the radius be $r$ then the red length on the vertical edge is given by $a - r$; similarly the blue length on the horizontal edge is given by $b - r$.

Using the property that tangential distances are equal we get $a - r + b - r = c$.

$\therefore a + b - 2r = c \Rightarrow r = \dfrac{a + b - c}{2}$

Why would this result be invalid if the triangle were not right angled?

Problem ID: 364 (03 Nov 2009)     Difficulty: 2 Star

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