Distance between the Orthocenter and Circumcenter

Problem 2 from the 1994 Asian Pacific Mathematical Olympiad asked to prove, for any triangle Δ, the inequality

where H is the orthocenter, O the circumcenter and R the circumradius of Δ.

A stronger result can be easily proved with complex numbers: for any triangle Δ, with side lengths a, b, c the following identity holds:

Solution

Set the origin of the complex plane at the circumcenter O of triangle Δ and let x, y, z be the complex number corresponding to the vertices of Δ opposite sides a, b, c. Then, as we know, the orthocenter H can be determined with

With these, (1) becomes

With a little insight and an appeal to symmetry, we are led to establishing an identity from which (2) follows immediately.

The left-hand side of (3) equals

The second term in the right-hand side of (3) equals

Together they make (3) obvious.

(The same identity is derived elsewhere from the relation between the medians and the side lengths of a triangle.)

Complex Numbers

1. Algebraic Structure of Complex Numbers
2. Division of Complex Numbers
3. Useful Identities Among Complex Numbers
4. Useful Inequalities Among Complex Numbers
5. Trigonometric Form of Complex Numbers
6. Real and Complex Products of Complex Numbers
7. Complex Numbers and Geometry
8. Remarks on the History of Complex Numbers
   • First Geometric Interpretation of Negative and Complex Numbers
9. Complex Numbers: an Interactive Gizmo
10. Cartesian Coordinate System
11. Fundamental Theorem of Algebra
12. Complex Number To a Complex Power May Be Real
13. One can't compare two complex numbers
14. Riemann Sphere and Möbius Transformation
15. Problems
    • Product of Diagonals in Regular N-gon
    • Sum of the Nth Roots of Unity
    • Napoleon's Relatives
    • Distance between the Orthocenter and Circumcenter

Copyright © 1996-2009 Alexander Bogomolny