Shirali, Shailesh
(2018)
How to prove it.
At Right Angles, 7 (2).
pp. 95100.
ISSN 25821873
Abstract
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 18thcentury 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.
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