Complex numbers and the Erdös - Mordell inequality

François Dubeau

Département de mathématiques Université de Sherbrooke 2500 Boulevard de l’Université Sherbrooke (Qc), Canada, J1K 2R1

francois.dubeau@usherbrooke.ca

Submitted to publication.

Key Words : Complex numbers, dot product, cross product, polar form, cartesian form, triangle inequality, Erdös-Mordell inequality.

MSC: Primary: 51M04, Secondary: 30A10, 15A72.

Summary

Let ΔABC be a triangle and O a point in it. Consider a coordinate system with O as its origin and A is on the positive direction of the X-axis, B is in the first or second quadrant, and C is in the third or fourth quadrant. We associate to the vertices A, B, and C their corresponding complex numbers

(0.1)  \begin{matrix}A = |A|e^{iθ_{A}} & , & θ_{A} = 0, \\ B = |B|e^{iθ_{B}} & , & θ_{B} ∈ (0, π), \\ C = |C|e^{iθ_{C}} & , & θ_{C} ∈ [π, 2π), & and & θ_{C} − θ_{B} ∈ (0, π). \end{matrix}

The conditions on the angles imply that O is in the triangle. We also associate to the foot of each perpendicular from O to the sides their corresponding complex number : P for the side joining the vertices B and C, Q for the side joining the vertices C and A, and R for the side joining the vertices A and B. With this notation, the Erdös-Mordell inequality is

(0.2)  2(|P | + |Q| + |R|) ≤ |A| + |B| + |C|.

In this short note we use dot and cross products of complex numbers, operations usually introduced in textbook as a curiosity, see [3, 4]. For the history of this inequality and references about other proofs, see [1].

References

[1] C. Alsina and R.B. Nelson, A visual proof of the Erdös-Mordell inequality, Forum Geometricorum, 7 (2007), 99-102.

[2] A. Avez, A short proof of a theorem of Erdös and Mordell, The Amererican Mathematical Monthly, 100 (1993), 60-62.

[3] E.B. Saff and A.D. Snider, Fundamental of Complex Analysis with Applications to Engineering and Science, Third Edition, Prentice Hall, New Jersey, 2003.

[4] M.R. Speigel, Theory and Problems of Complex Variables, Schaum’s outline series, McGraw-Hill, New York, 1964.