Determine if a triangle is right angled with only coordinates

Question

What is the easiest way to determine that, given 3 coords, they DON'T form a right angled triangle?

EG, (0, 0, 0), (0, 1, 0), (1, 0, 0) - forms a right angled triangle

(0, 0, 0), (0, 1, 0), (1, 0.5, 0) - does not form a right angled triangle

Answer 1

3 points always form a triangle.

For a triangle with side lengths $a$, $b$, $c$, the Pythagorean theorem states that if and only if $a^2 + b^2 = c^2$ then the triangle is a right triangle.

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If $a$ is the distance between points $p$ and $q$, with $p = \begin{bmatrix} p_1 \\ p_2 \\ p_3 \end{bmatrix}$, and $q = \begin{bmatrix} q_1 \\ q_2 \\ q_3 \end{bmatrix}$,

then $$a = |P - Q| = \sqrt{(p_1 - q_1)^2 + (p_2 - q_2)^2 + (p_3 - q_3)^2}$$

so

$$a^2 = (p_1 - q_1)^2 + (p_2 - q_2)^2 + (p_3 - q_3)^2$$

Answer 2

Draw it and test the angle that seems likely taking the dot product of the two vectors corresponding to the sides of that vertex. You can test all three if you don't want to draw.

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Alternatively calculate the side lengths, if $c$ is the largest and $c^2=a^2+b^2$ it is a right triangle, otherwise it isn't.

Answer 3

For this, I believe it is best to calculate the length of every side before using Pythagoras theorem, hope this helps!