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January 3, 2019

Understanding Simson Lines

Imagine a point P floating in the plane of a triangle ABC. How far is this point from the sides of the triangle?

A point P is floating in the plane of a triangle ABC

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 know theorem. Try to prove this using simple angle chasing.

Pedal Triangle
Pedal Triangle
P goes to circumference -> Pedal Triangle thins out to become a line

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