square recovery

Geometry Level 3

You are given a square and 4 random points on its sides (1 point on each of the 4 sides). Then the sides are erased and you are left with only the 4 points.

Is it possible to recover the original square? If so, how can one do it?

Assume that the 4 random points do not happen to form the vertices of square (as in that case there are infinite number of squares that could be the original one)

Yes, using both ruler and divider Yes, using only ruler Yes, using only divider No

Naukhatsky Gregory by Naukhatsky Gregory , 33, Russia

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1 solution

Let A,B,C,D be the random points. If we construct CX perpendicular to BD and with length equal to BD, then X would lay on the side (as triangles BLD and CKX are equal). Given point X one can easily construct all sides one by one.

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After you recover point X show the next steps diagramatically.

Vijay Simha - 2 years, 8 months ago

If the points really are random, then this will work 100% of the time. However, it's possible for this method not to work. For example if the points are (2,1), (1,2), (-2,1) and (1,-2). These point's lie on a square, but X coincides with A, so you can't construct the line AX, and the method doesn't work.

Joe Mansley - 1 year, 5 months ago

1 pending report

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