By Igor Pak

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**Extra info for Lectures on Discrete and Polyhedral Geometry**

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1] The aspect ratio ρ of a rectangle is the ratio of its sides. Prove that every centrally symmetric polygon Q ⊂ R3 contains an inscribed centrally symmetric rectangle with a given aspect ratio ρ > 0. 1. 4. [2] A quadrilateral A in R3 is called regular if all four sides are equal and all four angles are equal. Prove that every polygon in R3 has an inscribed regular quadrilateral. Generalize this to Rd , for all d ≥ 3. 5. ♦ [1] Let A be an inscribed quadrilateral into a polygon Q ⊂ R3 .

For every convex polygon Q ⊂ R2 there exist three concurrent lines which divide Q into six parts of equal area. 33 Proof. Fix a line ℓ ⊂ R2 and take ℓ1 ℓ which divides Q into two parts of equal area. 1). Move z continuously along ℓ1 . Observe that the angle α defined as in the figure, decreases from π to 0, while angle β increases from 0 to π. By continuity, there exists a unique point z such that r2 and r2′ form a line. Denote this line by ℓ2 . As in the previous proof, rotate ℓ continuously, by an angle of π.

A) [1+] Let Q1 , . . , Qm ⊂ R2 be convex polygons in the plane with weights w1 , . . , wm ∈ R (note that the weights can be negative). For a region B ⊂ R2 deﬁne the weighted area as w1 area(B ∩ Q1 ) + . . + wm area(B ∩ Qm ). Prove that there exist two orthogonal lines which divides the plane into four parts of equal weighted area. 2 to weighted areas. 6. [2-] Let Q be a convex polygon in the plane. A line is called a bisector if it divides Q into two parts of equal area. Suppose there exists a unique point z ∈ Q which lies on at least three bisectors (from above, there is at least one such point).

### Lectures on Discrete and Polyhedral Geometry by Igor Pak

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