Co-ordinates of the 3rd point in a triangle

Question

Given :

Point A (a1,b1)
Point B (a2,b2)
Distance between A and B
Distance between A and C
Angle between AB and AC = 90deg

Have to find : C(a3,b3)

I have the co-ordinates of 2 points A(x,y) and B(p,q) but want to find the co-ordinates of a third point C(m,n).
I know the distance between A and B, A and C, and the angle between A and C which is 90deg.
I know this is simple Pythagoras theorem. But how do I implement it in php and what would be the formula?

Answer 1

Let $x,$y and $p,$q be the given coordinates of A and B, furthermore call $d the known distance between A and C and $d0 the known distance between A and B. By doing a little math you get the following formulae (here I'm directly implementing it in PHP):

$m = $x + ($q - $y) * $d / $d0;
$n = $y - ($p - $x) * $d / $d0;

There is also a second solution:

$m = $x - ($q - $y) * $d / $d0;
$n = $y + ($p - $x) * $d / $d0;

EDIT: Here is how I got the equations: I rotated the vector AB, which has the coordinates ($p - $x, $q - $y), by 90 degrees to obtain ($q - $y, -($p - $x)) and (-($q - $y), $p - $x) (depending whether clockwise or counter-clockwise) and then got the vector AC by scaling it with $d / $d0 (the ratio of their lengths). Now I just translated the vector by ($x, $y) to get ($m, $n).

Maybe this can be implemented more elegantly by using a vector class in PHP or even a whole library, but I think for this simple calculation it is much easier to implement it "by hand".