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 1 of 6: Label the intersection boundary

Previous step Next step

C_1D_1C_2D_2\cdots C_nD_n

Detailed analysis

Because the intersection has 2n sides, sides inherited from S and T must alternate. Label the vertices clockwise so that $C_iD_i$ are blue boundary segments and $D_iC_{i+1}$ are red boundary segments, with indices cyclic.