Two equal sides of an isosceles triangle are along \( -x + 2y = 4 \) and \( x + y = 4 \). If \( m \) is the slope of its third side, then the sum of all possible distinct values of \( m \) is:
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Use trigonometric identities and the relationship between slopes to solve geometry-related problems involving lines and angles.
The slope of the third side is determined by the angle between two lines using the formula:
\[
\tan(\theta) = \frac{m - \frac{1}{2}}{1 + \frac{1}{2}m}
\]
Rearranging this equation for the third side's slope results in a quadratic equation for \( m \):
\[
2m^2 - 3m + 1 = m^2 + 3m + 2
\]
After simplification and solving for the sum of the roots, we find \( m_1 + m_2 = 6 \).
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