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Proof of Pick's Theorem

Age 16 to 18 Challenge Level:

poly with hole
We are allowed to assume that any polygon, convex or not, can be split into a finite number of non-overlapping triangles.

However in this proof we assume that the interior of the polygon does not have any holes like the red polygon shown with a yellow hole in the diagram. Pick's formula is related to Euler's formula and ${\rm area }(P) = i + {1\over 2}p - q$ where $q$ depends on the number of holes.