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April 3, 2019

Pappus Theorem - Story from Math Olympiad

Alexander the great founded Alexandria in Egypt. This was in the year 332 Before Christ. 

Since then Alexandria became one of the most important cities of the world. It had the Light House and the Great Library. These were the wonders of ancient world. 

Perhaps a much less known wonder was Pappus the geometer. Pappus perhaps lived in Alexandria about 600 after the city was founded along the coast of the Mediterranean Sea. And with him was born one of the wonders of modern mathematics: Projective geometry. 

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Lets take a straight line. In fact take another one. 

Suppose the points A, B and C are on the first one. And a, b, c (in small) are in the second one. 

Pause for a moment. We have not specified how far apart these points are. Neither did we say anything about the inclination of the two lines. In other words, we did not say a thing about length or angle information. 


Let us join Ab and aB. Suppose their intersection point is 1

Next join Ac and aC. Suppose their intersection point is 2

Finally join Bc and bC and their intersection point is 3. 

Do you notice anything in particular about the points 1, 2 and 3? 

Aha ... they are collinear ... that is on the same straight line. 

It is big deal to say three points are on the same straight line. Take any two points (from the three), and you can draw a straight line through them. Usually the third point may or may not lie on that straight line. 

In this case, however, all three of them line up nicely. 

So far so good? 

Now the magic begins! You may exchange the roles of lines and points in everything that we did so far.

Start with two points L1 and L2 (instead of two lines).

Next draw lines A, B, C through L1 and a, b, c through L2. 

Next join Ab (a point) and aB (a point) to create line 1

Similarly join Ac and aC to create line 2

And finally Bc and bC to create line 3

Voila! 1, 2, 3 these lines pass through a single point. 

This remarkable theorem shows that lines and points are not so different after all. They are in some sense dual to each other. 

In fact some thing more remarkable is under the hood. 

Instead taking two lines L1 and L2, take a circle. Now pick points A, B, C and a, b, c on the circle. 

Now join Ab and aB to find intersection point 1

Join Ac and aC to find intersection point 2 and Join Bc and bC to find intersection point 3. 

Do you notice anything? 1, 2, 3 points have lined up again in a straight. 

In fact this works even on a parabola.

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