Problem 1
Let \(\triangle A B C\) be an equilateral triangle with center \(O .\) Let \(\ell\) be the line through \(O\) perpendicular to the plane of \(\triangle A B C .\) Prove that there is a point \(D\) on \(\ell\) such that the triangles \(\triangle D A B, \triangle D B C, \triangle D C A,\) and \(\triangle A B C\) are all congruent. Calculate the length \(O D\) in terms of \(A B\)
Problem 2
Let \(A B C D\) be a square with center \(O .\) Let \(\ell\) be the line through \(O\) perpendicular to the plane of \(A B C D .\) Prove that there is a point \(E\) on \(\ell\) such that the four triangles \(\triangle E A B, \triangle E B C, \triangle E C D,\) and \(\triangle E D A\) are all equilateral. Calculate the length \(O D\) in terms of \(A B .\) How can you use this construction to construct a regular octohedron?
Problem 4
Given a Platonic solid \(P\) you can construct a new polyhedron whose vertices are the centers of the faces of \(P .\) This new polyhedron is called the dual of \(P\) and it turns out that it is also a Platonic solid. For each of the five types of Platonic solids, identify the dual.
Problem 5
Our proof of Euler's theorem involved "flattening" a polyhedron. Draw the flattened form of a regular tetrahedron, a cube, and a regular octahedron.
Problem 8
Prove that if \(A\) and \(B\) are lattice points such that there are no lattice points on \(\overline{A B}\) and \(C\) is a lattice point that minimizes the distance to \(\overleftrightarrow{A B},\) then \(\triangle A B C\) is fundamental.
Problem 10
Given \(a > c > 0\) and \(b > d > 0\) define \(\Delta A B C\) to be the triangle with vertices \(A=(0,0), B=(a, b)\) and \(C=(c, d) .\) Prove that \(\triangle A B C\) has area \(\frac{1}{2}(b c-a d) .\) Use this to find a point \(C\) such that \(\triangle A B C\) will be fundamental, for \(B=(5,3)\)
