MHB Determining the Degrees of an Angle Given Three X and Y Coordinates

[xyle]

How do I determine the degrees of an angle if I three X and Y coordinates? I honestly just need a formula to plug into some code. Thank you in advance.

 

[MarkFL]

How do I determine the degrees of an angle if I three X and Y coordinates? I honestly just need a formula to plug into some code. Thank you in advance.

Suppose the 3 points are given by:

$$\left(x_i,y_i\right)$$ where $$i\in\{1,2,3\}$$

Now further suppose we wish to make one line segment from point 1 to point 2, and another from point 2 to point 3, and then find the angle, in degrees, subtended by the two segments. I would begin by defining the vectors:

$$a=\left\langle x_2-x_1,y_2-y_1 \right\rangle$$

$$b=\left\langle x_3-x_2,y_3-y_2 \right\rangle$$

And then, from the dot product of the two vectors, we may write:

$$\theta=\frac{180^{\circ}}{\pi}\arccos\left(\frac{a\cdot b}{|a||b|}\right)$$

where:

$$a\cdot b=\left(x_2-x_1\right)\left(x_3-x_2\right)+\left(y_2-y_1\right)\left(y_3-y_2\right)$$

$$|a|=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$$

$$|b|=\sqrt{\left(x_3-x_2\right)^2+\left(y_3-y_2\right)^2}$$

Does that make sense?