MHB Minimal Possible Values of Triangle Point Distances - POTW #266 Jun 7th, 2017

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The discussion centers around finding the minimal possible values of the expression PX^2 + QX^2 + RX^2 for a point X within triangle PQR, where the sides are given as PQ=3, QR=4, and PR=5. The problem is part of the Problem of the Week (POTW) series. Kaliprasad is recognized for providing the correct solution. The suggested solution details the methodology used to arrive at the minimal value. This mathematical exploration emphasizes the geometric properties of triangles and point distances.
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Here is this week's POTW:

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Let $PQR$ be a triangle such that $PQ=3,\,QR=4$ and $PR=5$. Let $X$ be a point in the triangle. Find the minimal possible values of $PX^2+QX^2+RX^2$.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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Congratulations to kaliprasad for his correct solution, and you can find the suggested solution as follows::)

Let the perpendicular distance from $X$ to $QR$ and $QP$ be $x$ and $y$ respectively. Then we have

$\begin{align*}PX^2+QX^2+RX^2&=x^2+y^2+(3-x)^2+(4-y)^2+x^2+y^2\\&=3\left(y-\dfrac{4}{3}\right)^2+3(x-1)^2+\dfrac{50}{3}\end{align*}$

This suggests that the minimal possible value of $PX^2+QX^2+RX^2$ is $\dfrac{50}{3}$.
 
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