Problem 3

Let two equal regular

n

-gons

S

and

T

be located in the plane such that their intersection is a

2n

-gon (

n \ge 3

). The sides of

S

are colored red and the sides of

T

blue. Prove that the sum of the lengths of the blue sides of

S\cap T

equals the sum of the lengths of its red sides.

Step 2 of 6: Define the two boundary sums

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x:=\sum_{i=1}^{n}p_i,\qquad y:=\sum_{i=1}^{n}q_i,\qquad a:=\text{the side length of }S

Detailed analysis

Let $p_i$ be the blue side of the intersection boundary associated with the $i$ th vertex of $S$ , and $q_i$ the red side associated with the corresponding vertex of $T$ . Thus $x$ is the total blue length and $y$ the total red length. The intersection perimeter is $x+y$ , strictly less than the perimeter $na$ of either original polygon.