math - Find a tangent point on circle?

Question

Given a line with first end point P(x1,y1) another end point is unknown, intersect with a circle that located at origin with radius R at only one point(tangent) T(x2,y2). Anyone know how to get the point T? Thanks in advance!

Answer 1

Given a line with first end point P(x1,y1) another end point is unknown, intersect with a circle that located at origin with radius R at only one point(tangent) T(x2,y2). Anyone know how to get the point T?

Some of the other solutions seem a little like overkill. I think the simplest way is just to notice that this is a right triangle, with vertices P, T, and O (the origin). The angle PTO is the right angle, because a tangent line is always at a right angle to a radius.

You know the length of TO because it's of length r and has a vertex at the origin; you know OP because you know where O and P is. Given two sides of a right triangle, it's easy to find the length and direction of the third side. This is homework, so I'll leave the rest as an exercise to the reader.

                    __...------__    T(x2, y2)                      
               _.-''             -(+)
            ,-'                   |----             
          ,'                     |     ----
        ,'                      |       '  ----
       /                       |         `     ----       
      /                       |           `.       ----   
     /                       |                        ----
    |                       |               |              ----
    |                      |                 |                  ----
    |                     |                  |                      ----
    |                   (+)---------------------------------------------(+) P (x1,y1)
    |                                        .'        
    |                    O                   |         
     |                                      .'         
                                           /          
                                         ,'           
        `                                /             
         '.                            ,'              
           '-.                      _,'             
              '-._              _,(+)  T'(x3, y3)                   
                  '`--......---'                       

There are two possible directions for TO, since the point T' is also a valid tangent point, so you will have two congruent triangles.