Distance between the Orthocenter and Circumcenter

Problem 2 from the 1994 Asian Pacific Mathematical Olympiad asked to

Prove that, for any triangle Δ, the inequality

HO ≤ 3R,

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:

(1) HO ² = 9R ² - (a² + b² + c²).

Solution

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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

  1. | x | = | y | = | z | = R,
  2. H = x + y + z,
  3. a = | z - y |, b = | x - z |, c = | y - x |.

With these, (1) becomes

(2) | x + y + z |² = 9R ² - (| z - y |² + | x - z |² + | y - x |²).

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

| x + y + z |² = 3(| x |² + | y |² + | z |²) - (| z - y |² + | x - z |² + | y - x |²).

The left-hand side of (3) equals

| x + y + z |² = | x |² + | y |² + | z |² + (xy* + yz* + zx* + x*y + y*z + z*x).

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

| z - y |² + | x - z |² + | y - x |² = 2(| x |² + | y |² + | z |²) - (xy* + yz* + zx* + x*y + y*z + z*x).

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. Plane Isometries As Complex Functions
  9. Remarks on the History of Complex Numbers
  10. Complex Numbers: an Interactive Gizmo
  11. Cartesian Coordinate System
  12. Fundamental Theorem of Algebra
  13. Complex Number To a Complex Power May Be Real
  14. One can't compare two complex numbers
  15. Riemann Sphere and Möbius Transformation
  16. Problems

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Copyright © 1996-2017 Alexander Bogomolny

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