How long is the Sun closer to Earth then Mars is? A recent question at Space.se has generated a few comments that are interesting.  

But the answer to one particular comment to might address a lot of misconceptions. 


  I imagine the actual halfway point is close to the sun Source


As the two planets orbit the sun at different speeds, I assume it must be a significant amount.

How long is the Sun closer to Earth then Mars is?
Let us first consider the Earth a fixed point and Mars moving around the Sun on a circular orbit with angular velocity equal to the relative angular velocity ($\omega_\bigoplus - \omega_♂)$
The distance between Mars and the Earth can be described as the square root of $R_E^2+R_M^2-2R_ER_Mcos(\theta)$, and when this is equal to $R_E^2$:  $cos(\theta) = \frac{R_E^2-R_E^2+R_M^2}{2R_ER_M} = 0.75$, meaning this happens when the elongation of Mars is around 0.72 and -0.72 radians (40 and -40 degrees). For smaller angles than 40, the cosine is bigger making the distance smaller (same goes for angles between $-40$ and $0$). Therefore, Mars is closer for $80/360 = 22$ percent of the synodic period of Mars, and for the remaining 78% or around 600 days.