The attached diagram shows a Triangle $abc$ formed from the connection of points $M, I_1, I_2, I_3$ such that some $2D$ space is shared equally between the points.
I would like to find the maximum area of the triangle defined by the lines $a, b, c$.
From Heron's formula the Area, $\mathcal{A}$ of the triangle with lines $a,b$ and $c$ is
$$\mathcal{A} = \sqrt{s(s-a)(s-b)(s-c)}\,\, \text{for: } s = \frac{a+b+c}{2}$$
And on further expansion:
$$\mathcal{A} = \frac{\sqrt{2a^2(b^2+c^2)-(b^2-c^2)^2-a^4}}{4}$$
From the diagram shown, the sides of the triangle $a,b,c$ are obtained from $d_1, d_2, d_3, \omega_1, \omega_2, \omega_3$ thus:
$$a = \frac{1}{2}(\sqrt{\frac{d_1^2-2d_1d_{2}\cos(\omega_1-\omega_{2})+d_{2}^2}{1-\cos^2(\omega_1-\omega_{2})}})\\
b = \frac{1}{2}(\sqrt{\frac{d_2^2-2d_2d_{3}\cos(\omega_2-\omega_{3})+d_{3}^2}{1-\cos^2(\omega_2-\omega_{3})}})\\c = \sqrt{[\frac{[d_1\sin\omega_{2}-d_{2}\sin\omega_1]\sin(\omega_{3}-\omega_{2})-[d_{2}\sin\omega_{3}-d_{3}\sin\omega_{2}]\sin(\omega_{2}-\omega_{1})}{2\sin(\omega_{2}-\omega_{1})\sin(\omega_{3}-\omega_{2})}]^2 + [\frac{[d_1\cos\omega_{2}-d_{2}\cos\omega_1]\sin(\omega_{2}-\omega_{3})-[d_{2}\cos\omega_{3}-d_{3}\cos\omega_{2}]\sin(\omega_{1}-\omega_{2})}{2\sin(\omega_{1}-\omega_{2})\sin(\omega_{2}-\omega_{3})}]^2}$$
The problems:
Task 1: Maximze $\mathcal{A}(\omega_3)$ subject to the following constraints
a. $-180\leq \omega_1,\omega_2,\omega_3 \leq 180$
b. $\omega_3 > \omega_2 > \omega_1$
c. $0.4 \leq d1,d2,d3 \leq 1.6$
d. $0.4 \leq \sqrt{d_1^2+d_2^2-2d_1 d_2\cos(w_1-w_2)}, \sqrt{d_2^2+d_3^2-2d_2 d_3\cos(w_2-w_3)},\\ \sqrt{d_1^2+d_3^2-2d_1 d_3\cos(w_1-w_3)}, \leq 1.6$
Task 2: Maximze $\mathcal{A}(\omega_2, \omega_3)$ subject to the following constraints
a. $-180\leq \omega_1,\omega_2,\omega_3 \leq 180$
b. $\omega_3 > \omega_2 > \omega_1$
c. $0.4 \leq d1,d2,d3 \leq 1.6$
d. $0.4 \leq \sqrt{d_1^2+d_2^2-2d_1 d_2\cos(w_1-w_2)}, \sqrt{d_2^2+d_3^2-2d_2 d_3\cos(w_2-w_3)},\\ \sqrt{d_1^2+d_3^2-2d_1 d_3\cos(w_1-w_3)}, \leq 1.6$
I do not have much experience in optimisation, hence unsure of how to go about the problems