Imagine a point P floating in the plane of a triangle ABC. How far is this point from the sides of the triangle?
Think: Suppose ABC is a triangle. Specify three positive numbers: x, y, z (for example 2, 3, 4.9). Is it possible to find a point P which is x unit away from AB, y unit away from BC and z unit away from CA?
If we think of the triangle ABC as a reference structure, it is immediately useful to find the distance of P from each of the sides.
Drop perpendiculars from P on the sides (or extended sides) of ABC. Suppose \( A_1, B_1, C_1 \) are the feet of the perpendiculars from P on BC, CA, AB respectively.
The triangle \( A_1, B_1 C_1 \) is known as Pedal Triangle. (The position of a P at a specific instance is known as the pedal point).
As the point, P floats on the plane of the triangle ABC, what happens to the pedal triangle?
Theorem: Pedal triangle thins out (degenerate) into a line as P floats to the circumference of triangle ABC. That is \( A_1, B_1, C_1 \) are collinear if and only if P is on the circumference of triangle ABC.
This is a well