MATH 245 midterm post-mortem

The midterm had 6 questions: 3 proofs done in class (Cauchy-Schwarz, orthogonal projection theorem, and parallelotope volume theorem — these can all be found in my notes), a question about simplex centroids, a question about a “sum-of-squares”-minimizing point in the plane, and a question about angles between orthogonal complements (which no one claimed to have solved).

I’ll post the questions (#4 to #6, since as I said the rest were just proofs from class) here. Feel free to post your solutions, or any ideas. If I have time, I’ll post my solution to #4 below (although it took me a lot of algebra to prove uniqueness…)

1. [#4 on exam] Let $[a_0, \ldots, a_\ell]$ be an $\ell$-simplex with centroid $g$. For each $k$ with $0 \leq k \leq \ell$, define $g_k$ to be the centroid of the $(\ell-1)$-simplex $\zeta_k = [a_0, \ldots, a_{k-1}, a_{k+1}, \ldots, a_\ell]$.  Define also $L_k$ to be the line passing through $a_k$ and $g_k$. Show that the lines $L_k$ all intersect uniquely at the point $g$.
2. [#5 on exam] Given $a_1, \ldots, a_\ell \in \mathbb{R}^2$, find a point $x \in \mathbb{R}^2$ minimizing the sum $\sum\limits_{i=1}^\ell |x-a_i|^2$. Prove this point is unique.
3. [#6 on exam] Let $U, V$ be hyperspaces in $\mathbb{R}^n$. Prove that $\mathsf{angle}(U,V) = \mathsf{angle}(U^\perp, V^\perp)$.

Also, sorry about the horribly outdated Spherical Geometry section in my notes. There are so many diagrams in my notebook that I haven’t had the time to transplant into LaTeX. I’ll do it the minute I get some spare time. Alternatively, you can just look at Professor New’s notes