Prove that, for any triangle Δ, the inequality
HO ≤ 3R,
where H is the orthocenter, O the circumcenter and R the circumradius of Δ.
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
- Algebraic Structure of Complex Numbers
- Division of Complex Numbers
- Useful Identities Among Complex Numbers
- Useful Inequalities Among Complex Numbers
- Trigonometric Form of Complex Numbers
- Real and Complex Products of Complex Numbers
- Complex Numbers and Geometry
- Remarks on the History of Complex Numbers
- Complex Numbers: an Interactive Gizmo
- Cartesian Coordinate System
- Fundamental Theorem of Algebra
- Complex Number To a Complex Power May Be Real
- One can't compare two complex numbers
- Riemann Sphere and Möbius Transformation
- Problems
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Copyright © 1996-2012 Alexander Bogomolny
