Difference between revisions of "1992 IMO Problems/Problem 4"
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Let <math>d</math> be the distance from point <math>M</math> to point <math>R</math> such that the coordinates for <math>R</math> are <math>(m+d,-r)</math> and thus the coordinates for <math>Q</math> are <math>(m-d,-r)</math> | Let <math>d</math> be the distance from point <math>M</math> to point <math>R</math> such that the coordinates for <math>R</math> are <math>(m+d,-r)</math> and thus the coordinates for <math>Q</math> are <math>(m-d,-r)</math> | ||
− | Let points <math>S</math> and <math> | + | Let points <math>S</math>, <math>T</math>, and <math>U</math> be the points where lines <math>PQ</math>, <math>PR</math>, and <math>l</math> are tangent to circle <math>C</math> respectively. |
In the plane let <math>C</math> be a circle, <math>l</math> a line tangent to the circle <math>C</math>, and <math>M</math> a point on <math>l</math>. Find the locus of all points <math>P</math> with the following property: there exists two points <math>Q</math>, <math>R</math> on <math>l</math> such that <math>M</math> is the midpoint of <math>QR</math> and <math>C</math> is the inscribed circle of triangle <math>PQR</math>. | In the plane let <math>C</math> be a circle, <math>l</math> a line tangent to the circle <math>C</math>, and <math>M</math> a point on <math>l</math>. Find the locus of all points <math>P</math> with the following property: there exists two points <math>Q</math>, <math>R</math> on <math>l</math> such that <math>M</math> is the midpoint of <math>QR</math> and <math>C</math> is the inscribed circle of triangle <math>PQR</math>. | ||
{{alternate solutions}} | {{alternate solutions}} |
Revision as of 16:54, 12 November 2023
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
Let points , , and be the points where lines , , and are tangent to circle respectively.
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.