[Math] How to calculate triangle coordinates in cartesian plane

analytic geometrytrianglesvectors

My problem can be describe by following image:

enter image description here

I know coordinates of an example P point. Say, they are equal to (8,8). I also know the length of a, b and c sides of the triangle which are equal to 10. Now, how one can calculate the coordinates of ABC points?

SOLVED (11/3/2016)

Thanks to @EmilioNovati answer to my other question ("How to calculate $B(x_1,y_1)$ when $\alpha$ and $A(x_0,y_0)$ are known?"), I found a solution to my problem with coordinates of triangle vertices.

First lets look at following drawing:

enter image description here

Now, in order to calculate $A(x_A,y_A)$, $B(x_B,y_B)$ and $C(x_C,y_C)$, one can do:

$x_C = x_P + a*cos(\alpha)$,
$y_C = y_P + a*sin(\alpha)$;

$x_A = x_P + a*cos(\alpha+\beta)$,
$y_A = y_P + a*sin(\alpha+\beta)$;

$x_B = x_P + a*cos(\alpha+\beta+\gamma)$,
$y_B = y_P + a*sin(\alpha+\beta+\gamma)$.

Best Answer

Hint:

using elemental geometrical properties of equilateral triangles you can see that:

$$ PC=PA=PB=\frac{10\sqrt{3}}{3} $$

and the distance of $P$ from the sides is half this value.

Can you do from this adding or subtracting such values from the coordinates of $P$ to find the vertices?

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