I think following similar simple fact is also true:
Suppose X1X2X3X4X5X6 is one convex hexagon H such that:
X1X2 // X3X6 // X4X5
X2X3 // X4X1 // X5X6
X3X4 // X5X2 // X6X1
Suppose three lines X1X4, X2X5, X3X6 bound one triangle T.
The similar result:
Area(X1X3X5) = Area(X2X4X6) = Average of areas of H and T.

The hexagon X1X2X3X4X5X6 can be see and constructed as hexagon parallel to medial triangle of T in the page:
https://www.cut-the-knot.org/Curriculum/Geometry/MedialPursuit.shtml

in MathWorld, this hexagon is named as Thomsen's hexagon:
https://mathworld.wolfram.com/ThomsensFigure.html

Best regards,
Bui Quang Tuan

This is a specialization of the parpolygon problem. If the diagonals are parallel to the sides, the two small triangles simply coalesce into one.

>in MathWorld, this hexagon is named as Thomsen's hexagon:
>https://mathworld.wolfram.com/ThomsensFigure.html

I was unaware of the terminology. Many thanks.