Correlation? (looking for way to find single closest point)

In Statistics, I have learned to find a best-fitting regression line for a set of data points on a coordinate plane. However, I was wondering if there was a way to find a single point on the plane that is closest to the most points in the dataset.

For example, assume there is a map showing the locations of houses in a city which require the service of a hospital. Is there a mathematical way to determine where the hospital should be centrally located in order to be in close proximity to the most homes?

I wouldn't know where to begin on this situation. Is there perhaps an equation or something that can be used?

Mathematically, such a point would have coordinates that are the mean of the coordinates of the existing dataset. The only problem is that the mean is influenced by extreme values, so caution should be used. The median of the coordinates would be a fairly safe choice.