I was reading the book "Foundations of Hyperbolic manifolds" by J. Ratcliffe, where I found the cosine formula for a geodesic triangle on the sphere. The result is as follows:
If $x, y, z \in \mathbb{S}^2$ are non-collinear points, and we consider a geodesic triangle formed by these points as the vertices with side lengths $a, b,$ and $c$, and opposite angles $\alpha, \beta$, and $\gamma$, then $$\cos \alpha = \frac{\cos a - \cos b \cdot \cos c}{\sin b \sin c}.$$
Now, the proof goes as follows:
Consider geodesics $\sigma_1: \left[ 0, a \right] \rightarrow \mathbb{S}^2$, $\sigma_2: \left[ 0, b \right] \rightarrow \mathbb{S}^2$, and $\sigma_3: \left[ 0, c \right] \rightarrow \mathbb{S}^2$ such that $\sigma_1 \left( 0 \right) = \sigma_3 \left( c \right) = y, \sigma_2 \left( 0 \right) = \sigma_1 \left( a \right) = z$ and $\sigma_3 \left( 0 \right) = \sigma_2 \left( b \right) = x$. Let the angles be as follows: $\theta \left( \sigma_1' \left( 0 \right), - \sigma_2' \left( b \right) \right) = \alpha$, $\theta \left( \sigma_2' \left( 0 \right), - \sigma_3' \left( c \right) \right) = \beta$, and $\theta \left( \sigma_3' \left( 0 \right), - \sigma_1' \left( a \right) \right) = \gamma$. That is, we are in the situation as shown in the figure.
We now focus on the point $x$. We will assume that $\alpha$ is an acute angle (as given in the book). With this, we consider the vectors $x \times y$ and $z \times x$, where $\times$ denotes the cross product on $\mathbb{R}^3$. Since both these vectors are perpendicular to $x$ (by definition of cross-product), and $\sigma_3' \left( 0 \right)$ and $-\sigma_2' \left( b \right)$ are perpendicular to $x$ since they are tangent to the sphere, we have that all four vectors are coplanar.
Now, the author writes that we are in the following situation:
While it is understood that $\theta \left( \sigma_1' \left( 0 \right), x \times y \right) = \frac{\pi}{2}$, and $\theta \left( -\sigma_2' \left( b \right), z \times x \right) = \frac{\pi}{2}$, due to some geometric arguments, what is unclear to me is that why do $x \times y$ and $z \times x$ are on the "opposite sides"? That is, why can such a thing not happen?
Also, what happens if $\alpha$ is not an acute angle? Any insights into this are appreciated!