Fermat Points and Concurrent Euler Lines

Let F1 and F2 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, F1, F2 (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.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

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Copyright © 1996-2018 Alexander Bogomolny

In the applet,