title: How to solve for point perpenediculuar to intersection of two circles?
body: <p>I am working through <a href="http://paulbourke.net/geometry/circlesphere/" rel="nofollow noreferrer">Paul Bourke's formula to find the intersection of two circles</a> and I am stuck on the step to find "a". I will outline the relevant part of the formula.</p>
  
  <p>Consider two triangles: P<sub>0</sub>P<sub>2</sub>P<sub>3</sub> and P<sub>1</sub>P<sub>2</sub>P<sub>3</sub>.</p>
  
  <p>let P<sub>0</sub> and P<sub>1</sub> be the centers of the two circles. <br/>
  let P<sub>3</sub> be the intersections of the two circles. <br/>
  let P<sub>2</sub> be the intersection between the two intersections of the circles and the line between the two circles. <br/></p>
  
  <p>let d be the distance between the two circle centers = ||P<sub>1</sub> - P<sub>0</sub>|| = distance formula on the circle centers. <br/>
  let a be the distance from P<sub>0</sub> to P<sub>2</sub> and let b be the distance from P<sub>1</sub> to P<sub>2</sub>.</p>
  
  <p>Using the Pythagorean theorem we can deduce:</p>
  
  <blockquote>
    <p>a<sup>2</sup> + h<sup>2</sup> = r<sub>0</sub><sup>2</sup> and b<sup>2</sup> + h<sup>2</sup> = r<sub>1</sub><sup>2</sup> <br/></p>
  </blockquote>
  
  <p>Here is where I get tripped up:</p>
  
  <blockquote>
    <p>Using d = a + b we can solve for a, <br/>
           a = (r<sub>0</sub><sup>2</sup> - r<sub>1</sub><sup>2</sup> + d<sup>2</sup> ) / (2 * d) <br/></p>
  </blockquote>
  
  <p>I do not understand where the formula to find a comes from. Can someone outline how this is deduced?</p>
  
  <p>Thank you!</p>