Optimising Miquel circles centre triangles and third pedal triangles

This paper presents two, seemingly new, phenomena of Euclidean geometry. In the first part, we discuss two different proofs that the Miquel circles centre triangle has minimal area if, and only if, the line segments joining the Miquel point M with the points D,E,F on the triangle sides or their extensions are perpendicular to the triangle sides. In the second part, we maximise the area of the third pedal triangle which, of course, varies with the initial point P; the area of the third pedal triangle turns out to be a maximum if, and only if, the point P coincides with the incentre of the triangle. So both parts involve optimisation which is an important field of mathematics, since mathematicians and other people have been fascinated by finding maxima and minima for centuries.