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Ghost Quadrilateral


Saad drew a quadrilateral $ABCD$ such that $AB = BC = CD$. He also drew the midpoints $M, N$ and $P$ of the sides $AB, BC$ and $CD$ respectively. But then Rafi came and erased everything but the midpoints. Figure out a way to reconstruct the quadrilateral using the points $M, N$ and $P$.



Proof Based Problems  


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Solution

Disclaimer: The solutions we've shared are just one exciting approach, and there are surely many other wonderful methods out there. We’d love to hear your alternative solutions in the community thread below, so let's keep the creativity flowing!

We start by connecting $M$,$N$ and $N$,$P$. Let the perpendicular bisectors of $MN$ and $NP$ intersect at $Q$. $F$,$E$ are the midpoints of $MN$ and $NP$ respectively.


Let $Q'$ be the reflection of $Q$ across $N$.

Construct parallelogram $Q'BQC$ such that $B\in QF$ and $C\in QE$.

Let $A$ and $D$ be the reflections of $B$ and $C$ across $M$ and $P$.


$\square ABCD$ is the quadrilateral Saad had drawn.

This is a proof based problem added for learning purposes and does not accept submissions.

You can view the solution by clicking on the solution tab.

Editorial


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