Pedal Triangle and Isogonal Conjugacy: What is this about?
A Mathematical Droodle

Explanation

The first assertion is an immediate consequence of the definition of the pedal triangle: the vertices of the pedal triangle of point P are the feet of perpendiculars from P to the sides of ABC. Naturally then the perpendiculars from P to the sides of ABC are concurrent: they meet in P. By Maxwell's theorem, the perpendiculars from the vertices of ABC onto the sides of A'B'C' are also concurrent.

To see that point P' of concurrency is the isogonal conjugate of P, observe that in quadrilateral AB'PC' the angles at B' and C' are right. The quadrilateral is therefore cyclic. Which implies that angles APC' and AB'C' are equal (as inscribed angles subtending the same arc.) Let X be the foot of perpendicular from A to B'C' (that passes through P'.) It follows that the right triangles APC' and AB'X are similar and their other acute angles are also equal:

C'AP = XAB' = P'AB',

so that AP and AP' are isogonal to each other. The same consideration applies to the vertices B and C, thus proving the second assertion.

(The first property is mentioned in [Honsberger, p. 64]. Both properties are offered as exercises in [TCCT, p. 215].)

References

1. R. Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, MAA, 1995, pp. 153-154
2. C. Kimberling, Triangle Centers and Central Triangles, Congressus Numerantium 129, Utilitas Mathematica Publishing, 1998.

1. Gergonne and Medial Triangles Are Orthologic
2. Pedal Triangle and Isogonal Conjugacy
3. Orthologic Triangles in a Quadrilateral