===== Теорема =====
$$I_aI=\frac{r}{\sin \frac{\beta}{2} \cdot \sin \frac{\gamma}{2}} = \sqrt{r^2+r^2_a+(p-b)^2+(p-c)^2}$$
$$I_aI_b=r \cdot \frac{\operatorname{ctg} \frac{\alpha}{2}+\operatorname{ctg} \frac{\beta}{2}}{\sin \frac{\gamma}{2}} = \dfrac{r\cos{\frac{\gamma}{2}}}{\sin{\frac{\alpha}{2}}\sin{\frac{\beta}{2}}\sin{\frac{\gamma}{2}}}=\dfrac{r\ctg{\frac{\gamma}{2}}}{\sin{\frac{\alpha}{2}}\sin{\frac{\beta}{2}}}$$

$$\sin \frac{\alpha}{2}=\sqrt{\frac{(p-b)(p-c)}{b c}}$$
$$\operatorname{ctg} \frac{\alpha}{2}=\frac{p-a}{r}$$
$$I_aI=a \sqrt{\frac{b c}{p(p-a)}} = \sqrt{(b-c)^2+(r+r_a)^2}$$
$$I_{a}I_{b}=c \sqrt{\frac{a b}{(p-a)(p-b)}} = \sqrt{c^2+(r_a+r_b)^2}$$