On propellers from triangles (Q5948430)

The propeller from triangles configuration has a triangle \(\Delta ABC\) as the shaft and three triangles \(\Delta APX\), \(\Delta YBQ\), \(\Delta RZC\) as blades. Let \(U,V,W\) be the midpoints of the lines \(YR\), \(ZP\), and \(XQ\) respectively. A well-known result is that the shaft and blades being equilateral triangles forces the triangle \(\Delta UVW\) to be equilateral. In [\textit{M. Gardner}, Coll. Math. J. 30, 18-22 (1999)] it is shown geometrically that if the shaft and the blades are directly similar, then the triangle \(\Delta UVW\) is also directly similar to the shaft \(\Delta ABC\). The author gives an algebraic proof of this result using complex numbers. He also investigates other questions of how relations among the shaft and blades of the propeller affect the triangle \(\Delta UVW\). For example, if the shaft and blades are isocentric, then the triangle \(\Delta UVW\) is isocentric to the shaft \(\Delta ABC\). The reader is referred to the article for a complete description of the results.