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