Quadrilaterals!

Geometry Level 2

Once, while casually discussing Quadrilaterals, a dispute arose between two friends, A and B.
A: Once, I saw a square formed from the midpoints of the sides of another quadrilateral. And the interesting thing is that the outer quadrilateral wasn't a square!!

B: Aha! I caught you! It's impossible to make a square from the midpoints of the sides of another quadrilateral until and unless the outer quadrilateral is a square. I can prove it.

A: I can prove my point too.

B: So, let us go to my home, and see you lose.

They both went to B's home and resolved their confusion. Who must have won the argument?

A B

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Yatin Khanna by Yatin Khanna , 20, India

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2 solutions

Michael Mendrin
Sep 12, 2016

This shows how A can do this and win.

4 Helpful 0 Interesting 0 Brilliant 0 Confused

What kind of a quadrilateral is this, sir!
Till now, I thought only few kites and squares can make a square if we join the midpoints...

Edit: I deduced the necessary conditions. For the midpoints of a quadrilateral to form a square, its diagonals should be equal and perpendicular to each other.

Yatin Khanna - 4 years, 9 months ago

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If we draw 3 parallel lines that are equidistant from each other, as here passing through the top, middle, and bottom vertices of the upright square (or diamond), then any line passing through the 3 parallel lines is divided in half. Hence, we know that this quadrilateral, when the midpoints of its sides are joined, will form a perfect square (or diamond).

Michael Mendrin - 4 years, 9 months ago

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Thats a completely different way of thinking. WOW!!
Then, is my statement (about necessary conditions) false or is itnautomatically satisfied in this scenario?

Yatin Khanna - 4 years, 9 months ago

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@Yatin Khanna – I have no idea. Let me think about that.

Edit: Yeah, they're always perpendicular and of equal lengths, so your statement is not incorrect.

Michael Mendrin - 4 years, 9 months ago
Yatin Khanna
Sep 12, 2016

Any kite (that isn't a square) with equal diagonals will do the trick, not only squares.
Consider this:

Assumptions in my solution: Quad formed by joining midpoints of any quad is a parallelogram.

Edit: Any quadrilateral with equal diagonals that are perpendicular to each other will do the trick.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice question and solution. This was the example I found as well.

Brian Charlesworth - 4 years, 9 months ago

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