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