Chordal: Points with equal tangents to two circles
- Ryan Hirst
Given two circles, find the points whose tangents to both circles are equal.
- The Secant-Tangent Theorem: http://www.geogebratube.org/material/show/id/28302
- The locus line is often called the Chordal of circles A and B. ? Yeah, me either. I will call it the Power Line, Tangentipede, or Tangent Monster of A and B. Must consult the manual on Plastic Brontosaurus Algebra.
- Circles α, β have equal power at all points on the line. However, if the circles intersect, part of the locus line is a chord. The power at these points is negative, and it is not possible to draw tangents from them. Hence, the locus of points with equal tangents to two circles is all points on the power line whose power is positive (... which are outside both circles).
- The construction is partly backward: the radius of β is given by the blue circle, and I make up the perpendicular CF. These are not magic answers. They are tools I use to nose around the structure of a problem. I don't know how other people do it. Here, we bump into the solution right away by working backward. This is rarely the case. For Malfatti's Problem, for example, the circles in the first worksheet do not obey the constraints: http://www.geogebratube.org/material/show/id/32079. I figured out how to enforce them only when I found the answer to the problem. It is never necessary to know about an answer in advance, to solve a problem.