Three Way Split
Take any point P inside an equilateral triangle. Draw PA, PB and PC from the point P perpendicular to the sides of the triangle where A, B and C are points on the sides. Prove that the sum of the lengths PA + PB + PC is a constant.
Suppose there is an election with only 3 parties. Draw a diagram on which you can mark a point to show the percentage of winning candidates from each party. (This could be used on the night of the election, moving the point as the results for the different seats come in, showing how well each of the parties are doing overall). Shade the regions where one of the parties has a clear majority and a region where there is no overall majority.
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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
Triangle Midpoints
You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?
Floored
A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?