1992 IMO Problems/Problem 4
Problem
In the plane let be a circle, a line tangent to the circle , and a point on . Find the locus of all points with the following property: there exists two points , on such that is the midpoint of and is the inscribed circle of triangle .
Video Solution
https://www.youtube.com/watch?v=ObCzaZwujGw
Solution
Note: This is an alternate method to what it is shown on the video. This alternate method is too long and too intensive in solving algebraic equations. A lot of steps have been shortened in this solution. The solution in the video provides a much faster solution,
Let be the radius of the circle .
We define a cartesian coordinate system in two dimensions with the circle center at and circle equation to be
We define the line by the equation , with point at a distance from the tangent and cartesian coordinates
Let be the distance from point to point such that the coordinates for are and thus the coordinates for are
In the plane let be a circle, a line tangent to the circle , and a point on . Find the locus of all points with the following property: there exists two points , on such that is the midpoint of and is the inscribed circle of triangle .
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.