Challenge Level

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.