Question 135
Suppose $A$, $B$, $C$ are three non-collinear points corresponding to complex numbers $z_0 = ai$, $z_1 = \frac{1}{2} + bi$, $z_2 = 1 + ci$ ($a$, $b$ and $c$ being real numbers), respectively. Prove that the curve
$$
z = z_0 \cos^4 t + 2z_1 \cos^2 t \cdot \sin^2 t + z_2 \sin^4 t \ (t \in \mathbb{R})
$$
shares a single common point with the line bisecting $AB$ and parallel to $AC$ in $\triangle ABC$, and find this point.
$$
z = z_0 \cos^4 t + 2z_1 \cos^2 t \cdot \sin^2 t + z_2 \sin^4 t \ (t \in \mathbb{R})
$$
shares a single common point with the line bisecting $AB$ and parallel to $AC$ in $\triangle ABC$, and find this point.