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Difference between revisions of "1992 IMO Problems/Problem 4"

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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.
 
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.
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First we get the coordinates for points <math>S</math> and <math>T</math>.
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Since the circle is the incenter we know the following properties:
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<math>\left| RU \right| = \left| RT \right|=(m+d)</math>
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and
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<math>\left| QU \right| = \left| QS \right|=(m-d)</math>
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Therefore, to get the coordinates of point <math>T=(T_{x},T_{y})</math>, we solve the following equations:
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<math>T_{x}^{2}+T_{y}^2=r^{2}</math>
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<math>\left| RT \right|^{2}=(m+d-T_{x})^{2}+(r+T_{y})^2</math>
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<math>(m+d)^{2}=(m+d-T_{x})^{2}+(r+T_{y})^2</math>
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After a lot of algebra, this solves to:
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<math>T_{x}=\frac{2r^{2}(m+d)}{(m+d)^{2}+r^{2}}</math>
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<math>T_{y}=\frac{r\left[ (m+d)^{2}-r^{2} \right]}{(m+d)^{2}+r^{2} }</math>
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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 17:05, 12 November 2023

Problem

In the plane let $C$ be a circle, $l$ a line tangent to the circle $C$, and $M$ a point on $l$. Find the locus of all points $P$ with the following property: there exists two points $Q$, $R$ on $l$ such that $M$ is the midpoint of $QR$ and $C$ is the inscribed circle of triangle $PQR$.

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 $r$ be the radius of the circle $C$.

We define a cartesian coordinate system in two dimensions with the circle center at $(0,0)$ and circle equation to be $x^{2}+y{2}=r^{2}$

We define the line $l$ by the equation $y=-r$, with point $M$ at a distance $m$ from the tangent and cartesian coordinates $(m,-r)$

Let $d$ be the distance from point $M$ to point $R$ such that the coordinates for $R$ are $(m+d,-r)$ and thus the coordinates for $Q$ are $(m-d,-r)$

Let points $S$, $T$, and $U$ be the points where lines $PQ$, $PR$, and $l$ are tangent to circle $C$ respectively.

First we get the coordinates for points $S$ and $T$.

Since the circle is the incenter we know the following properties:

$\left| RU \right| = \left| RT \right|=(m+d)$

and

$\left| QU \right| = \left| QS \right|=(m-d)$

Therefore, to get the coordinates of point $T=(T_{x},T_{y})$, we solve the following equations:

$T_{x}^{2}+T_{y}^2=r^{2}$

$\left| RT \right|^{2}=(m+d-T_{x})^{2}+(r+T_{y})^2$

$(m+d)^{2}=(m+d-T_{x})^{2}+(r+T_{y})^2$

After a lot of algebra, this solves to:

$T_{x}=\frac{2r^{2}(m+d)}{(m+d)^{2}+r^{2}}$

$T_{y}=\frac{r\left[ (m+d)^{2}-r^{2} \right]}{(m+d)^{2}+r^{2} }$






In the plane let $C$ be a circle, $l$ a line tangent to the circle $C$, and $M$ a point on $l$. Find the locus of all points $P$ with the following property: there exists two points $Q$, $R$ on $l$ such that $M$ is the midpoint of $QR$ and $C$ is the inscribed circle of triangle $PQR$.

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

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