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Just Opposite

A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

Fitting In

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ

Square Areas

Can you work out the area of the inner square and give an explanation of how you did it?

LOGO Challenge - the Logic of LOGO

Age 11 to 16
Challenge Level

One way to start:
What are the basic elements of the design?
Can you make each of these?
How is the design built up from these elements?
Can you put the elements together in a meaningful way? Are there some intermediate steps?

I don't think that a single chunk of code that no one can understand is elegant or helpful.

Perhaps you should not go for minimising the number of procedures but rather making clear what you have done to an audience.