INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More 

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

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com
enter