= Solution
Take the <trine state ensemble>
$$
\boxed{|t_j\rangle=\frac{|0\rangle+\omega^j|1\rangle}{\sqrt2},\qquad
p_j=\frac13,\qquad j=0,1,2,\quad\omega=e^{2\pi i/3}.}
$$
All three states are normalized. Since $1+\omega+\omega^2=0$, their average <density matrix> is
$$
\frac13\sum_{j=0}^2|t_j\rangle\langle t_j|
=\frac16\sum_{j=0}^2\begin{pmatrix}1&\omega^{-j}\\\omega^j&1\end{pmatrix}
=\frac I2.
$$
Geometrically, their unit <Bloch vectors> form an <equilateral triangle> on the equator and sum to zero. <Orthogonality> is unnecessary for an ensemble decomposition of a <mixed state>: three distinct, nonorthogonal <pure states> can give the same maximally mixed average as two orthogonal states.
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