By Descartes R.
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Suppose now that U ◦ and V ◦ do not intersect. 16). e. y and z lie close to each other on X). Clearly, for such (y, z), either both corresponding points u and v lie inside X or both u and v lie outside of X. Let A be the former and let C be the latter regions. From above, for all (y, z) ∈ B we have u ∈ / X and v ∈ X. In other words, when y is fixed and z is moved along X counterclockwise starting at y, of the points u and v the first to move outside of X is always u. Now, consider the smallest right equilateral triangle R inscribed into X (the existence was shown earlier).
We say that an equilateral triangle is inscribed into X if there exist three distinct points y1 , y2 , y3 ∈ X such that |y1 y2 | = |y1y3 | = |y2 y3 |. 2. For every simple polygon X = [x1 . . xn ] ⊂ R2 and a point z ∈ X in the interior of an edge in X, there exists an equilateral triangle (y1 y2 z) inscribed into X. The same holds for every vertex z = xi with ∠ xi−1 xi xi+1 ≥ π/3. First proof. Denote by X ′ the clockwise rotation of X around z by an angle π/3. Clearly, area(X) = area(X ′ ). Thus for every z in the interior of an edge in X, polygons X and X ′ intersect at z and at least one other point v.
Now us consider inscribed rhombi with one diagonal parallel to a given line. We start with a special case of convex polygons. 9. Every convex polygon in the plane has an inscribed rhombus with a diagonal parallel to a given line. 5). g. 19). First proof. Let X = ∂A be a convex polygon in the plane and let ℓ be a given line. For a point x ∈ A X, denote by a1 , a2 the intersections of a line though x and parallel to ℓ with X. 11). Let f : A → A be a function defined by f (x) = cm{a1 , a2 , b1 , b2 }, where cm{·} denotes the center of mass.