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 4 of 6: Compare the equal perimeters

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(h+k)x+y=(h+k)y+x\implies (h+k-1)x=(h+k-1)y

Detailed analysis

On each side of $S$ , the two adjacent corner-triangle pieces contribute $kp_i$ and $hp_{i+1}$ , while the intervening red boundary piece contributes $q_i$ . Summing gives $\operatorname{per}(S)=(h+k)x+y$ . Similarly, $\operatorname{per}(T)=(h+k)y+x$ . Since $S$ and $T$ are congruent, their perimeters are equal, which yields the displayed relation.