# 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 two Fermat's points and one of the vertices A, B, C. There are three such triangles. The other triangles are considered separately.

What if applet does not run? |

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Copyright © 1996-2018 Alexander Bogomolny

In the applet,

- A'B'C' is the outer and a'b'c' the inner Napoleon's triangles,
- F and f are the corresponding Fermat's points,
- G and G' are the centroids of ΔABC and ΔFfC, respectively,
- O is the circumcenter of ΔFfC,
- P the intersection of B'C' with a'c',
- M is the midpoint of AB, m the midpoint of Ff,
- F' the reflection of f in M.
We only prove the claim for ΔFfC.

What if applet does not run? As we know, the two Napoleon triangles share the centroid G. Both are invariant under the rotations through 120° with center G. One such rotation maps A'B' and a'b' onto B'C' and c'a', respectively. Therefore, it also maps O to P, implying that

∠OGP = 120°. Since∠Oa'P = 120°, the four points O, G, a', P are concyclic. Their circumcircle also contains B' because∠PB'O = 60°. Therefore,(1)

∠OGB' = ∠Oa'B'.

The same rotation maps ∠Oa'B' onto ∠Pc'C', showing that the two are equal:

(2)

∠Oa'B' = ∠Pc'C'.

Since Pc'⊥Bf and C'c'⊥ BA,

(3)

∠Pc'C' = ∠fBA.

Further, since ∠BF'A = ∠AFB = 120° = 180° - ∠AfB, the quadrilateral AfBF' is cyclic, giving ∠fBA = ∠fF'A. It follows that

(4)

∠fF'A = ∠OGB'.

Now, from AF' || fB⊥ A'C' and B'G⊥ A'C', we see that

AF' || B'G. This, together with (4), yields(5)

F'f || GO.

Points G and G' divide the segments CM and Cm in ratio 2:1, causing

GG' || Mm. By construction,FM = MF' and, since alsoFm = mf, the same argument shows thatMm || F'f. Combining this with (5) shows thatGG' || GO, implying the collinearity of G', G and O.### References

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Copyright © 1996-2018 Alexander Bogomolny