Euler line proof

The famous Euler line is named after Leonhard Euler who in 1765 proved the following remarkable theorem, which was not known to the ancient Greeks.

Euler Line Theorem
The orthocenter (H), centroid (G) & circumcentre (O) of any triangle are collinear (lie in a straight line), and HG = 2GO.

Challenge
a) Can you explain why (prove that) the result is true?
b) Can you prove it in more than one way?

 

Euler Line Proof

Proof
While the result can be proved in several different ways, and proofs appear in many classic texts such as Coxeter & Greitzer (1967), the two-step proof below using transformation geometry, is probably the shortest & most elegant way to prove it.
1) The midpoint △A'B'C' is homothetic to △ABC (i.e. similar to and corresponding sides parallel) with scale factor (1/2).
2) The half-turn of H around G and its subsequent dilation with factor (1/2) maps it to H'. But H' is the circumcentre O of the original △ABC. Therefore, H, G & O (= H') of any triangle are collinear, and HG = 2GO.

Illustration
i) Click on the 'Show Rotation' button to show △ABC shaded in yellow.
ii) Drag the point X on the circle counter-clockwise to rotate the yellow triangle until it is rotated through 180^o (a half-turn) around G. (Observe where the image H' of H is moving as you rotate the yellow triangle).
iii) After rotating the yellow triangle through 180^o, click on the 'Show Dilation' button to show a red triangle covering the yellow one.
iv) In the slider at the bottom now drag point Y to the right to dilate the red triangle by the scale factor a/b until it becomes 1/2. (Observe where the dilated image H'' of H' is moving as you dilate the red triangle).
v) In iv) above you should now have seen how the dilated image H'' of H' has moved onto O, and completes the proof.

Related Links
The Center of Gravity of a Triangle (Rethinking Proof activity)
Point Mass (Vertex) Centroid of Quadrilateral
Homothetic Polygons Concurrency
Triangle Centroids of a Hexagon form a Parallelo-Hexagon (A generalization of Varignon's Theorem)
Nine Point Conic and Generalization of Euler Line
A further generalization of the Euler line
Generalizations of a theorem of Sylvester (about the Euler line)
Euler-Nagel line Analogy
Euler and Nagel lines for Cyclic and Circumscribed Quadrilaterals
The quasi-Euler line of a quadrilateral and a hexagon
Concurrency and Euler line locus result
Four Collinear Points
Triangle Incentre-Circumcentre Collinearity
Collinear Conjecture
Investigating incentres of some iterated triangles
Investigating circumcentres of iterated median triangles
Concurrency, collinearity and other properties of a particular hexagon
Quadrilateral Similar Triangles Collinearity (A generalization of a collinearity theorem for 'equilic' quadrilaterals)
An interesting collinearity (An application of Desargues & Pappus)
Generalizations involving maltitudes
Pirate Treasure Hunt and a Generalization
A variation of Miquel's theorem and its generalization
A Forgotten Similarity Theorem?

References
Coxeter, H.S.M. & Greitzer, S.L. (1967). Geometry Revisited. Washington, DC: Math. Assoc. Amer.
Honsberger, R. (1995). Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer.
Kelly, Adam. (2020). Geometry Revisited – Before Transformations.

External Links
Euler line (Wikipedia)
Leonhard Euler (Wikipedia)
Euler Line (Encyclopedia of Triangle Centers)
Euler line (Wolfram MathWorld)
Euler line (Art of Problem Solving MathWorld)
The Euler Line and the 9-Point Circle (Cut The Knot)
Spiral similarity (Wikipedia)
Homothety (Wikipedia)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

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