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 3 of 6: Relate the similar triangle sides

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\frac{a_i}{p_i}=h,\quad\frac{b_i}{p_i}=k,\quad\frac{c_i}{q_i}=k,\quad\frac{d_i}{q_i}=h

Detailed analysis

Let the triangle at the vertex associated with the blue side $p_i$ have the other side lengths $a_i,b_i$ , and let the triangle associated with the red side $q_i$ have other side lengths $c_i,d_i$ . Similarity around the alternating boundary gives constants $h,k$ with $a_i=hp_i$ , $b_i=kp_i$ ; the opposite orientation of the second family gives $c_i=kq_i$ , $d_i=hq_i$ .