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'Thebault's Theorem' printed from http://nrich.maths.org/

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Why do this problem ?

It gives an opportunity for experimentation and making your own conjecture, which then needs to be proved. Three different methods of proof are based on coordinates, vectors and complex numbers.

Key questions

Four points are vertices of a parallelogram. What do you know about the line segments joining them?

If on a line segment you draw a square:
  • what do you know about the vertices of the square?
  • can you find the coordinates, or the position vector of the centre of the square?

Possible support

The problem Napolean's Theorem is similar and perhaps easier.