Find the incentre of the triangle formed by $(0, 0), (2, 2)$ and $(1 - \sqrt{3}, 1 + \sqrt{3})$

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Solution

$a = \sqrt{(1 - \sqrt{3} - 2)^{2} + (1 + \sqrt{3} - 2)^{2}} = \sqrt{(1 + \sqrt{3})^{2} + (1 - \sqrt{3})^{2}} = \sqrt{1 + 3 + 2 \sqrt{3} + 1 + 3 - 2 \sqrt{3}} = 2 \sqrt{2} b = \sqrt{(1 - \sqrt{3})^{2} + (1 + \sqrt{3})^{2}} = 2 \sqrt{2} c = \sqrt{2^{2} + 2^{2}} = 2 \sqrt{2} a s t r i a n g l e i s a n e q u i l a t e r a l t r i a n g l e a n d h e n c e i t s c e n t r o i d w i l l b e i t s i n c e n t r e I = ((\frac{0 + 2 + 1 - \sqrt{3}}{2}), (\frac{0 + 2 + 1 + \sqrt{3}}{2})) = ((\frac{3 - \sqrt{3}}{2}), (\frac{3 + \sqrt{3}}{2}))$

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I. The centroid of the triangle formed by $(2, 3, - 1), (5, 6, 3), (2, - 3, 1)$ is	a) $(2, 2, 2)$
II. The circumcentre of the triangle formed by $(1, 2, 3), (2, 3, 1), (3, 1, 2)$ is	b) $(3, 1, 4)$
III. The orthocenter of the triangle formed by $(2, 1, 5), (3, 2, 3), (4, 0, 4)$ is	c) $(1, 1, 0)$
IV. The incentre of the triangle formed by $(0, 0, 0), (3, 0, 0), (0, 4, 0)$ is	d) $(3, 2, 1)$
	e) $(0, 0, 0)$

y =∣ x ∣

y = 1

\frac{x}{a} + \frac{y}{b} = 1

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