A little fun with the Erdős-Mordell Inequality Theorem

One of my favorite subjects in mathematics is Geometry.  You would think that, after 2,000 years or so, we would know all there is to know about the subject.  Of course, you would be horribly wrong.  Take, for example, this fun theorem that usually isn’t mentioned in a high school Geometry course.  Originally proposed by the legendary Paul Erdős in 1935, it was later proven by in 1937 by Louis Mordell and D. F. Barrow.

Erdos-MordellTheorem

The Erdős-Mordell Inequality Theorem:  If P is a point inside of ΔABC, and PA, PB and Pare the feet of the perpendiculars from P upon the respective sides BC, CA, and AB, then

 PA+PB+PC>=2(PP_A+PP_B+PP_C).

Here’s the fun part, can you prove it?  It doesn’t hurt to start with a simpler case.  Consider an equilateral triangle where P is the circumcenter.  Consider an isosceles triangle where P is the circumcenter.  Can you prove it for these simpler cases?  Yes?  Good.  Now, can you prove it for the general case?  It isn’t as easy as it looks.

Note: diagram and Inequality provided at :Weisstein, Eric W. “Erdős-Mordell Theorem.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Erdos-MordellTheorem.html

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The unique personality of a young mathematician.

A young genius ...

A young genius …

One of the all time great mathematicians was Paul Erdős.  Erdős published more original mathematics than any other mathematician in history.  However, Erdős was more than a mathematician.  He was also one of the most unique personalities in all of mathematics.  This “uniqueness” started at a young age.

Upon meeting a new “friend”, Erdős often introduced himself in a mathematical way.  Sometimes he would ask “how many ways can you prove the Pythagorean Theorem?”  (Erdős himself knew 37 different proofs by his early teens!)  Other times, he would ask a computational question.  Once, when 17, he was introduced to 14-year-old Andrew Vazsonyi.  Immediately, without any greeting, he asked Vazsonyi to give him a four digit number.  Without blinking, Erdős was able to square the number in his head.  However, he apologized for not being able to cube the number.  As he said, “I am getting old and decrepit and cannot tell you the cube.”  Amazingly, by the age of 17, he already viewed himself as an old man who was losing his mathematical talents.  In fact, this obsession lasted all 83 years of his life.  Fortunately, for the mathematical community, this obsession never came to be.  He produced original mathematics up until the day he died.

If you would like to learn more about Paul Erdős, you can click here to get his biography.

If you would like to read some great books about Paul Erdős, you can look for one of these:

  • The Man Who Loved Only Numbers by Paul Hoffman  (Click here to read my brief synopsis.)
  • My Brain is Open by Bruce Schechter  (Click here to read my brief synopsis.)

These are great books about one of the greatest talents and most compassionate human beings in the mathematical world.  You will not be disappointed.