Reference: Complex Numbers from A to …Z by Titu Andreescu and Dorin Andrica
Balkan Mathematical Olympiad, 1985.
Let O be the circumcenter of the triangle ABC, let D be the mid-point of the segment AB, and let E be the centroid of the triangle ACD. Prove that lines CD and OE are perpendicular if and only if .
Let O be the origin of the complex plane and let a, b, c, d, e be the coordinates of points A, B, C, D, E respectively. Then,
Using the real product of complex numbers, if R is the circumradius of triangle ABC, then
Lines CD and DE are perpendicular if and only if . That is,
The last relation is equivalent to
, that is, — call this equation I.
On the other hand, is equivalent to . That is,
or, , hence, —- equation II
The relations (1) and (2) show that CD is perpendicular to OE, if and only if .
Ref: Complex Numbers from A to Z by Titu Andreescu and Dorin Andrica
Thanks Prof Andreescu!