How to find a point at a certain distance to other points on the same line

Question

Assuming the points A(x1,y1) and B(x2,y2) and distances between AB (d1) and AC (d2) are known. How can I find the point C(xp,yp)?

Actually it has a trivial solution, writing the distance equation 2 times between AB and AC and also getting the slope for the line AB, and substituting it back gives the solution. However, it is also possible for my case that the slope can be infinity and also I do not want to get two possible answers due to squareroot operation and I need to derive an equation that is valid for all cases. How can one achieve that?

Thanks

Answer 1

$x_p=x_1\pm \dfrac{d_2}{d_1}(x_2-x_1)$

and

$y_p=y_1\pm \dfrac{d_2}{d_1}(y_2-y_1)$

Of course you need to take the same sign for the two equations. You have only $2$ solutions, not $4$.

EDIT: if you want the opposite side, you have to choose the $-$ sign.

EDIT2: I was not awake... There is no $\sqrt{}$...