# Circles Connected to a Triangle

1) If $r$ denotes in-radius, then

$r = \sqrt {\frac{{(s – a)(s – b)(s – c)}}{s}} = \frac{\Delta }{s}$
$r = \frac{{a{\text{ }}Sin\frac{\beta }{2}Sin\frac{\gamma }{2}}}{{Cos\frac{\alpha }{2}}} = \frac{{{\text{b }}Sin\frac{\gamma }{2}Sin\frac{\alpha }{2}}}{{Cos\frac{\beta }{2}}} = \frac{{{\text{c }}Sin\frac{\alpha }{2}Sin\frac{\beta }{2}}}{{Cos\frac{\gamma }{2}}}$
$r = (s – a)Tan\frac{\alpha }{2} = (s – b)Tan\frac{\beta }{2} = (s – c)Tan\frac{\gamma }{2}$

2) The circum radius $R$ is given by

$R = \frac{a}{{2Sin\alpha }} = \frac{b}{{2Sin\beta }} = \frac{c}{{2Sin\gamma }} = \frac{{abc}}{{4\Delta }}$

3) If ${r_1},{r_2},{r_3}$ denotes $e –$radii, then

${r_1} = \frac{\Delta }{{s – a}}$
${r_1} = (s – b)Cot\frac{\gamma }{2} = (s – c)Cot\frac{\beta }{2} = a\frac{{Cos\frac{\beta }{2}Cos\frac{\gamma }{2}}}{{Cos\frac{\alpha }{2}}}$
${r_2} = \frac{\Delta }{{s – b}}$
${r_2} = (s – c)Cot\frac{\alpha }{2} = (s – a)Cot\frac{\gamma }{2} = b\frac{{Cos\frac{\alpha }{2}Cos\frac{\gamma }{2}}}{{Cos\frac{\beta }{2}}}$
${r_3} = \frac{\Delta }{{s – c}}$
${r_3} = (s – a)Cot\frac{\beta }{2} = (s – b)Cot\frac{\alpha }{2} = c\frac{{Cos\frac{\alpha }{2}Cos\frac{\beta }{2}}}{{Cos\frac{\gamma }{2}}}$

4) ${r_1} + {r_2} + {r_3} – r = 4R$

5) In an equilateral triangle $r:R:{r_1} = 1:2:3$