Can the intersections of the unit sphere with two distinct planes be the same?

Question

Let $A_1, A_2, B_1, B_2, C_1, C_2, D_1, D_2 \in \mathbb{R}$ be such that $A_i^2 + B_i^2 + C_i^2 \neq 0$, $i \in \{1,2\}$. Define $\Pi_i := \big\{(x,y,z)\in\mathbb{R}^3 :\ A_ix + B_iy + C_iz + D = 0\big\}$, $i \in \{1,2\}$. Then due to the conditions on $A_i$, $B_i$, and $C_i$, $\Pi_1$ and $\Pi_2$ are planes in $\mathbb{R}^3$. Denote by $\Sigma$ the unit sphere in $\mathbb{R}^3$, i.e. define $\Sigma := \big\{(x,y,z)\in\mathbb{R}^3 :\ x^2+y^2+z^2=1\big\}$. Denote by $Z_i$ the intersection of $\Sigma$ with $\Pi_i$, $i \in \{1,2\}$. Suppose that $Z_1 = Z_2 \neq \emptyset$. Is it necessarily the case that $\Pi_1 = \Pi_2$? In other words, is there necessarily some scalar $s \in \mathbb{R}$ such that the following equations hold simultaneously? $$ \begin{align*} A_2 &= sA_1\\ B_2 &= sB_1\\ C_2 &= sC_1\\ D_2 &= sD_1 \end{align*} $$