How to calculate triangle coordinates in cartesian plane?

My problem can be describe by following image:

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:

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

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

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

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

• Is $AB$ parallel to the $x$ axis? Also, is $P$ the centroid? – mathlove Mar 11 '16 at 17:18
• Yes, $AB$ is parallel to the $x$ axis, and also $P$ is the centroid. – bluevoxel Mar 11 '16 at 17:25

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?

• elemental?! wow :) – Zubin Mukerjee Mar 11 '16 at 17:09
• Dont'you know that if $l$ is the side of an equilateral triangle the height $h$ is $h=l\sqrt{3}/2$? and the the three heights intersect at a point $P$ such that the distance form a vertex is double the distance from a side? See: en.wikipedia.org/wiki/Equilateral_triangle#Principal_properties – Emilio Novati Mar 11 '16 at 17:13
• Yes, I know. I am just more familiar with the use of "elementary" as opposed to "elemental" in this context. But I looked up the definition and yours is perfectly fine. – Zubin Mukerjee Mar 11 '16 at 17:18
• Sorry! My english is so poor :( – Emilio Novati Mar 11 '16 at 17:22
• @EmilioNovati I believe that I should use trigonometry here, but I don't know how can I use angles values in calculation of ABC coordinates. – bluevoxel Mar 11 '16 at 18:14