A Proof from my Book

This is proof from my book - my proof of my all-time favorite true result of nature - Pick's Theorem. This is the simplest proof I have seen without using any high pieces of machinery like Euler number as used in The Proofs from the Book.

Given a simple polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon vertices are grid points, Pick's theorem provides a simple formula for calculating the area A of this polygon in terms of the number i of lattice points in the interior located in the polygon and the number b of lattice points on the boundary placed on the polygon's perimeter:

In the language of pictures,

Play around with this in Geogebra.

Kudos to some of the Students of Cheenta Ganit Kendra like Shahbaz Khan and ACHUTHKRISHNAN, who have successfully completed the proof with the help of the following stepstones:

Steps:

Step 1:

Consider the two-dimensional plane. What is the minimum area of a triangle whose vertices are points with integer coordinates?

Hint: Use Determinant form to compute the area of such a triangle.

Such triangles are called Fundamental Triangles.

Step 2

Prove that the triangle with minimum area and with coordinates as integers cannot contain ANY integer point on or inside the triangle except the three vertices.

Hint: Prove by contradiction.

Observe that the above picture is one such example.

Step 3

Find the area of a triangle with integer vertices with m points inside it and n points on its boundary edges. (in terms of m and n ) if possible otherwise prove that it is not possible.

Hint: Just draw pictures and use the previous steps.

Step 4

Find the area of any polygon with integer vertices with m points inside it and n points on its boundary edges. (in terms of m and n ) if possible otherwise prove that it is not possible.

Hint: Just try to follow the ideas in Step 3. Try to observe by introducing one point and by the method of induction.

I would love if you try the steps and discover the Pick's Theorem in such simple steps yourself. Don't forget to try out the interactive Geogebra version as given.

Share your views and ideas in the comments' section.

Enjoy. 🙂