在圆上找一个切点? [英] Find a tangent point on circle?

问题描述

给定一条具有第一个端点 P(x1,y1) 的线，另一个端点未知，与位于原点的圆相交，半径为 R，仅在一个点 (切线) T(x2,y2) 处相交.有谁知道如何获得T点?提前致谢！

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!

推荐答案

给定一条具有第一个端点 P(x1,y1) 的线，另一个端点未知，与位于原点的圆相交，半径为 R，仅在一个点 (切线) T(x2,y2) 处相交.有谁知道如何获得T点?

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?

其他一些解决方案似乎有点矫枉过正.我认为最简单的方法就是注意这是一个直角三角形，有顶点 P、T 和 O(原点).角度 PTO 是直角，因为切线始终与半径成直角.

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.

你知道 TO 的长度，因为它的长度是 r 并且在原点有一个顶点；你知道 OP 因为你知道 O 和 P 在哪里.给定直角三角形的两条边，很容易找到第三条边的长度和方向.这是家庭作业，所以我将把剩下的留给读者作为练习.

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)                   
                  '`--......---'                       

TO 有两个可能的方向，因为点 T' 也是一个有效的切点，所以你会有两个全等的三角形.

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

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