Proof: Pi is Constant
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, defined as the ratio of a circle's circumference to its diameter, is constant. That is, it does not vary given circles of differing size. This claim is equivalent to saying "all circles are proportional" or "all circles are similar."
PrerequisitesEdit
 Laws of similar triangles
 Laws of limits
ProofEdit
Let Circle_{1} represent a circle of radius r_{1}, and construct it. Let Circle_{2} represent a circle of radius r_{2}, such that r_{2} > r_{1} and Circle_{2} is concentric with Circle_{1}, and construct it. Choose an integer and let . Construct an isosceles triangle, , such that each equal leg has length r_{2}, the vertex adjacent to both equal legs (v_{2}) lies at the center of Circle_{2}, and the measure of the angle at v_{2} is equal to . Let s_{2} represent the length of the side opposite . By this, it can be seen that filling Circle_{2}'s interior with n triangles congruent to such that all share a vertex at the center of Circle_{2}, the equal legs extend outward from that vertex, and none overlap, a regular polygon is created whose perimeter approximates the circumference of Circle_{2}.
By the above constructions, a similar isosceles triangle, , is made such that each equal leg has length r_{1}, the vertex adjacent to both equal legs (v_{1}) lies at the center of Circle_{1}, and the measure of the angle at v_{1} is equal to . Let s_{1} represent the length of the side opposite in . Let c_{2} represent the circumference of Circle_{2}, and c_{1} represent the circumference of Circle_{1}. Then:
Further, let n increase without bound. Then:
and

(the ratio of Circle_{2}'s circumference to its diameter) 

(the ratio of Circle_{1}'s circumference to its diameter) 
To prove the proposition, it suffices to show that:
By the laws of similar triangles:
Substituting:
QED