# Fermat Points and Concurrent Euler Lines

Let F_{1} and F_{2} denote the (inner and outer) *Fermat-Toricelli* points of a given ΔABC. We prove that the Euler lines of the 10 triangles with vertices chosen from A, B, C, F_{1}, F_{2} (three at a time) are concurrent at the centroid of ΔABC [Beluhov]. This is obviously true for ΔABC itself. The applet below illustrates the case where the triangles are formed by one of the Fermat points and a pair of vertices A, B, C. There are six such triangles. The other three triangles are considered separately.

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Suffice it to consider one of the six triangles. We will actually consider two at once, viz., ΔBCF, where F is one of the Fermat points.

In any triangle, the Euler line passes through the centroid, the circumcenter, the orthocenter and the nine-point center. The circumcircle of ΔBCF, coincides with the circumcircle of the equilateral ΔBCA'', implying that A', the center of ΔBCA'', is the circumcenter of ΔBCF. What we are going to show is that the line GA' passes through the centroid, say, G' of ΔBCAF.

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Let M_{a} be the midpoint of side BC. Observe that _{a}:CG = 1:2_{a}:A'A'' = 1:2_{a} and divides it in the same ratio: G'M_{A}:FG' = 1:2. On the other hand, FM_{a} is cut by GA' in the ratio 1:2. It follows that G' lie at the intersection of GA' and FM_{a} and, in particular, on GA' making it the Euler line of ΔBCF.

### References

### Napoleon's Theorem

- Napoleon's Theorem
- A proof with complex numbers
- A second proof with complex numbers
- A third proof with complex numbers
- Napoleon's Theorem, Two Simple Proofs
- Napoleon's Theorem via Inscribed Angles
- A Generalization
- Douglas' Generalization
- Napoleon's Propeller
- Napoleon's Theorem by Plane Tessellation
- Fermat's point
- Kiepert's theorem
- Lean Napoleon's Triangles
- Napoleon's Theorem by Transformation
- Napoleon's Theorem via Two Rotations
- Napoleon on Hinges
- Napoleon on Hinges in GeoGebra
- Napoleon's Relatives
- Napoleon-Barlotti Theorem
- Some Properties of Napoleon's Configuration
- Fermat Points and Concurrent Euler Lines
- Fermat Points and Concurrent Euler Lines II
- Escher's Theorem
- Circle Chains on Napoleon Triangles
- Napoleon's Theorem by Vectors and Trigonometry
- An Extra Triple of Equilateral Triangles for Napoleon
- Joined Common Chords of Napoleon's Circumcircles
- Napoleon's Hexagon
- Fermat's Hexagon
- Lighthouse at Fermat Points
- Midpoint Reciprocity in Napoleon's Configuration
- Another Equilateral Triangle in Napoleon's Configuration
- Yet Another Analytic Proof of Napoleon's Theorem
- Leo Giugiuc's Proof of Napoleon's Theorem
- Gregoie Nicollier's Proof of Napoleon's Theorem
- Fermat Point Several Times Over

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