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Take a Square

Cut off three right angled isosceles triangles to produce a pentagon. With two lines, cut the pentagon into three parts which can be rearranged into another square.

Three Way Split

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

* A'B'C' is an equilateral triangle. For any point P inside it, the area of * A'B'C' is equal to the sum of the areas of the triangles PB'C' , PC'A' and PA'B'

PA, PB and PC are the heights of triangles PB'C', PC'A' and PA'B'. If the length of the side of the equilateral triangle is L units, then:

Area * A'B'C' = ½ L x (PA + PB + PC)

PA + PB + PC = (2 x Area * A'B'C')/L = ½L * 3 = constant.

Figure 1
For each position of the point P inside the equilateral triangle we can take PA, PB and PC to represent the percentages of something which is split into three parts, the total being represented by PA +PB +PC . As P moves closer to A' the length PA increases and when P coincides with A' we have PB = PC = 0. We can label the vertices of the triangle to represent the three parts so that PA gives the percentage of A', PB gives the percentage of B', and PC gives the percentage of C'. Suppose, for example, these percentages are 60%, 30% and 10% respectively. If we draw an equilateral triangle with sides of length L = 200/ * 3 then PA + PB + PC = 100.
To find the position of P draw lines parallel to the sides of the triangle as shown in the diagram such that anywhere on the 60% line the perpendicular distance between this line and the line B'C' = 60 so PA = 60, similarly the distance between the 30% line and the line C'A' is 30 units so that PB = 30, and the distance between the 10% line and the line A'B' is 10 units so that PC = 10. Because PA + PB + PC = 100 (as we have shown) these three lines intersect in a single point. Figure 2
We apply this to the percentage of votes given to three parties A', B' and C' in an election (where there are no other candidates.) Points L, M and N are the midpoints of the sides of * A'B'C'.

If the point P lies on NM then height PA is half the height of * A'B'C' and PA represents exactly half the votes cast for party A'. If P is inside * A'NM then party A' has more than half the votes (a clear majority).

Similarly, if P is inside * B'LN then party B' has more than half the votes (a clear majority) and if P is inside * C'ML then party C' has more than half the votes (a clear majority). If P is inside * LNM then no party has more than half the votes (no overall majority).

Figure 3