Let $\triangle ABC$ and $\triangle A'B'C'$ be two triangles. The line through $A$ and perpendicular to $AA'$ meets the line through $B'$ and perpendicular to $BB'$ at $A_b$; The line through $A$ and perpendicular to $AA'$ meets the line through $C'$ and perpendicular to $CC'$ at $A_c$. Define $B_c, B_a, C_a, C_b$ cyclically then six points $A_b$, $A_c$, $B_c$, $B_a$, $C_a$, $C_b$ lie on a circle if only if $ABC$ and $A'B'C'$ have the same centroid.

This circle have a property: The main diagonals are equal: $A_bB_a=B_cC_b=C_aA_c$

[![enter image description here][1]][1]

**Question:** Is this result know?

This result is a generalization of the [Floor van Lamoen circle](https://mathworld.wolfram.com/vanLamoenCircle.html): Six circumcenters lie on a circle, and also is a generalization of [Elias M Hagos's six orthocenters lie on a circle](https://groups.io/g/euclid/message/3088)

**See also:** 

* [Geogebra applet](https://www.geogebra.org/m/zxqkdsbh)

* [Geogebra another](https://www.geogebra.org/m/k752gsev)

* [The paper](https://ijgeometry.com/wp-content/uploads/2014/10/9.pdf)


  [1]: https://i.sstatic.net/z0Mgq.png