  We extend the result of Yan to Broucke's isosceles orbit with masses $m_1$,
$m_1$, and $m_2$ with $2m_1 + m_2 = 3$. Under suitable changes of variables,
isolated binary collisions between the two mass $m_1$ particles are
regularizable. We analytically extend a method of Roberts to perform linear
stability analysis in this setting. Linear stability is reduced to computing
three entries of a $4 \times 4$ matrix related to the monodromy matrix.
Additionally, it is shown that the four-degrees-of-freedom setting has a
two-degrees-of-freedom invariant set, and linear stability results in the
subset comes `for free' from the calculation in the full space. The final
numerical analysis shows that the four-degrees-of-freedom orbit is linearly
unstable except for the interval $0.555 < m_1 < 0.730$, whereas the
two-degrees-of-freedom orbit is linearly stable for a much wider interval.
