How to prove it

Euler’s formula for the area of a pedal triangle Given a triangle ABC and a point P in the plane of ABC (note that P does not have to lie within the triangle), the pedal triangle of P with respect to △ ABC is the triangle whose vertices are the feet of the perpendiculars drawn from P to the sides of ABC. See Figure 1. The pedal triangle relates in a natural way to the parent triangle, and we may wonder whether there is a convenient formula giving the area of the pedal triangle in terms of the parameters of the parent triangle. The great 18th-century mathematician Euler found just such a formula (given in Box 1). It is a compact and pleasing result, and it expresses the area of the pedal triangle in terms of the radius R of the circumcircle of △ ABC and the distance between P and the centre O of the circumcircle