MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Mathematical Properties 49
Figure 2.27: Six Triangles [183]
Drawing three additional lines through the selected point which are parallel
to the sides of the original triangle partitions it into three parallelograms and
three small equilateral triangles (Figure 2.27 right). Since the areas of the
parallelograms are bisected by their diagonals and the equilateral triangles by
their altitudes,
A + C + E = x + a + y + b + z + c = B + D + F.
Figure 2.28: Pompeiu’s Theorem [184]
Property 36 (Pompeiu’s Theorem). If P is an arbitrary point in an equilateral
triangle ABC then there exists a triangle with sides of length PA, PB,
PC [184].
Draw segments PL, PM, PN parallel to the sides of the triangle (Figure
2.28). Then, the trapezoids PMAN, PNBL, PLCM are isosceles and thus
have equal diagonals. Hence, PA = MN, PB=LN, PC = LM and ∆LMN
is the required triangle. Note that the theorem remains valid for any point P
in the plane of ∆ABC [184] and that the triangle is degenerate if and only if
P lies on the circumcircle of ∆ABC [267].