# Lectures on discrete and polyhedral geometry by Pak I.

By Pak I.

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Exercises. 1. a) [1-] Prove that every simple polygon X ⊂ R2 and an interior point O there exist points x, y ∈ X such that O is a midpoint of (x, y). b) [1] Let X ⊂ R2 be a convex polygon with the center of mass cm(X) = O. Prove that X has an inscribed parallelogram with the center of mass at O. c) [1-] Let X = [x1 . . xn ] ⊂ R2 be a non-convex simple polygon. Prove that there exists a vertex xi and points y, z ∈ X such that xi is a midpoint of (y, z). Check that this does not necessarily hold for all vertices xi in the relative interior of the convex hull of X.

Formalize explicitly what it means to be generic. b) [1] Prove that there are 12 more squares with vertices on lines ℓi . c) [1] Compute the number of rectangles with given aspect ratio (ratio of its sides), such that all vertices lie on diﬀerent lines. d) [1] Let Q ⊂ R2 be a simple quadrilateral with diﬀerent edge lengths. Compute the number of quadrilaterals similar to Q, with all vertices on lines ℓi . 18. e. coplanar) rhombus. b) [1-] Prove that every ﬂat rhombus inscribed into a sphere is a square.

At the end, we have k − 1 pirates who make two cuts each, giving the total of 2(k − 1) cuts. 5. Exercises. 1. [1-] Let f : R → R be a periodic continuous piecewise linear function. Prove that f has inscribed chords of any length. / {1, 12 , 13 , . 2. ♦ [1+] For every α ∈ with f (0) = f (1) = 0, and such that α ∈ / D(f ). 5 cannot be extended to other values. 3. 5, prove that for every integer n there are at least n inscribed chords whose lengths are multiples of 1/n. 4. a) [1-] Consider two convex polygons in the plane.