Nina Guindilla has brought a beautiful proposition about pedal triangles. First, a definition:
Given a triangle and a point, the pedal triangle is obtained by projecting the given point onto the sides of the given triangle: the vertices of the pedal triangle are the feet of the perpendiculars from the given point to the sides of the given triangle. (Sometimes the pedal triangle degenerates and collapses to a line...)
Proposition. The pedal triangle of the pedal triangle of the pedal triangle is similar to the initial triangle (if no one degenerates).
Nina Guindilla proved the proposition with an inner point to the given triangle.
The three perpendicular segments from the point to the sides of the initial triangle divide this into three cyclic quadrilaterals (inscriptible in circles). Note now that angles with the same color are equal because intercept the same circle arc...
And observe below the dance of conguent angles: the small triangle is similar to the great one... QED.