Understand proofs concerning angles on circles

Question

I am reading the paper "On zeros of convex combination of polynomials" by Fell. In proving a theorem, the author listed two lemmas (Lemma 2 and Lemma 3) without proof.

How do we prove the two lemmas?

Answer 1

I'd prove the first one by observing that we can assume (by coordinate change) that $c$ is the origin, and by rotation, that $p$ is the point $(s, 0)$ on the $x$-axis, with $s > 1$. Further, we know that $a = (\cos \alpha, \sin \alpha)$ and $b = (\cos \beta, \sin \beta)$ for some angles $\alpha$ and $\beta$. Clearly angle $APB$ will be maximal if $\alpha$ and $\beta$ have opposite signs and are between $0$ and $\pi/2$.

At this point, I'd compute the angle $APB$ --- call it $u$ --- in terms of $\alpha$ and $\beta$, with a little trig. Then I'd set $$ \frac{\partial u(\alpha, \beta)}{\partial \alpha} = \frac{\partial u(\alpha, \beta)}{\partial \beta} = 0 $$ and solve them to show that in fact $\alpha = \beta$.

For the second, I'd again place the circle at the origin, and $a$ and $b$ at opposite angles $\pm \alpha$, so that the segment $ab$ is vertical. Then $p$ and $p'$ are $(\pm 1, 0)$, and the angles to be summed are computable from $\alpha$ and a little trig. Differentiate the sum with respect to $\alpha$ and set to $0$, and discover that the max occurs when $\cos(\alpha) = 0$, so that $a$ and $b$ are opposite ends of the vertical diameter.