MathLabs

Problem 3

Let two equal regular nn-gons SS and TT be located in the plane such that their intersection is a 2n2n-gon (n≥3n \ge 3). The sides of SS are colored red and the sides of TT blue. Prove that the sum of the lengths of the blue sides of S∩TS\cap T equals the sum of the lengths of its red sides.
Step 5 of 6: Factor the equation
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x(na−x)=y(na−y)  ⟹  (x−y)(na−x−y)=0.x(na-x)=y(na-y)\implies (x-y)(na-x-y)=0.
Detailed analysis

Cross-multiplication and rearrangement produce the two factors shown.