Let $A_1 A_2 \dotsb A_{11}$ be a regular 11-gon inscribed in a circle of radius 2. Let $P$ be a point, such that the distance from $P$ to the center of the circle is 3. Find
$PA_1^2 + PA_2^2 + \dots + PA_{11}^2.$

Question

Let $A_1 A_2 \dotsb A_{11}$ be a regular 11-gon inscribed in a circle of radius 2. Let $P$ be a point, such that the distance from $P$ to the center of the circle is 3. Find
$PA_1^2 + PA_2^2 + \dots + PA_{11}^2.$

anonymous · Answer

What exactly do you mean by the quantity $XY$ if both $X$ and $Y$ are points?

zmudz · Answer

@SithsAndGiggles there is no quantity XY. XY is a length of the segment formed by the points X and Y

anonymous · Answer

Okay, so assuming each of the $A_i$ are also points (which seems a reasonable assumption since they seem to denote the vertices of the $11$-gon), what do you mean by $P{A_i}^2$?

(By the way, a length is still a quantity)

zmudz · Answer

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Sorry for my terrible, simplified (in that I didn't draw an 11-gon) interpretation, @SithsAndGiggles , $PA_1 ^2$ would be the square of the distance between the point P and the point $A_i$. I've drawn the line segments in the diagram. P is going to be a point that is a distance of 3 units away from the center of the polygon, which is represented by O.

anonymous · Answer

Since the circle has radius $2$, naturally the point $P$, since it is a distance of $3$ away from the center, will lie a little bit outside the circle (and hence the $11$-gon).
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