Koostame sirgete võrrandid:

KL: $$y = kx + b$$ $$3 = -k + b$$ $$-3 = -2k + b$$ $$3 + k = -3 + 2k$$ $$6 = k$$ $$b=9$$

LM: $$y = kx + b$$ $$-3 = -2k + b$$ $$1 = 5k + b$$ $$-3 + 2k = 1 - 5k$$ $$7k = 4$$ $$k = \frac{4}{7}$$ $$b = -\frac{13}{7}$$ $$y = \frac{4}{7}x - \frac{13}{7}$$

MK: $$y=kx+b$$ $$1 = 5k + b$$ $$3 = -k + b$$ $$1 - 5k = 3 + k$$ $$6k = -2$$ $$k = -\frac{1}{3}$$ $$b = \frac{8}{3}$$ $$y = -\frac{1}{3}x + \frac{8}{3}$$

Leiame antud lõikude pikkused ning kolmnurga ümbermõõt ja pindala:

$$\vec{KL} = L - K = (-2, -3) - (-1, 3) = (-1, -6)$$ $$\vec{LM} = M - L = (5, 1) - (-2, -3) = (7, 4)$$ $$\vec{MK} = K - M = (-1, 3) - (5, 1) = (-6, 2)$$ $$\vert{\vec{KL}}\vert = \sqrt{(-1)^2 + (-6)^2} = \sqrt{37}$$ $$\vert{\vec{LM}}\vert = \sqrt{7^2 + 4^2} = \sqrt{65}$$ $$\vert{\vec{MK}}\vert = \sqrt{(-6)^2 + 2^2} = \sqrt{40}$$ $$P = \vert{\vec{KL}}\vert + \vert{\vec{LM}}\vert + \vert{\vec{MK}}\vert = \sqrt{37} + \sqrt{65} + \sqrt{40}$$ $$S = \sqrt{p(p-a)(p-b)(p-c)}$$ $$p = \frac{a + b + c}{2}$$ $$p = \frac{\sqrt{37} + \sqrt{65} + \sqrt{40}}{2}$$ $$S = 2\sqrt{5}$$

Koostame kolmnurga ümberringjoone võrrandi:

$$\begin{vmatrix} x^2 + y^2 & x & y & 1 \\ -1 & 3 & 1 & 1 \\ -2 & -3 & 1 & 1 \\ 5 & 1 & 1 & 1 \\ \end{vmatrix} = 0$$ $$x^2 + y^2 - x + 3y - 1 = 0$$

Arvutame ümberringi pindala, mis ei ole kolmnurga KLM poolt hõivatud:

$$\pi r^2 - S = \pi 5^2 - 2\sqrt{5} = 25\pi - 2\sqrt{5}$$