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> | ||
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+ | Let points <math>S</math> and <math>T</math> points where lines <math>PQ</math> and <math>PR</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 17:52, 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
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.