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LOGO Challenge 5 - Patch

Using LOGO, can you construct elegant procedures that will draw this family of 'floor coverings'?

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LOGO Challenge - Triangles-squares-stars

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

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LOGO Challenge - Tilings

Three examples of particular tilings of the plane, namely those where - NOT all corners of the tile are vertices of the tiling. You might like to produce an elegant program to replicate one or all of these.

Napoleon's Theorem

Stage: 4 and 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

You might like to make a tessellation with copies of $\Delta ABC$ and three triangles drawn on the sides of $\Delta ABC$ and prove your conjecture from the tessellation. This uses only elementary geometry. An alternative method is to use the Cosine Rule.

Alternatively you can use vectors or complex numbers.