The Airport  Problem

 

Let no one ignorant of geometry enter here.
-Plato( Inscription above Plato's academy )
 

 

There is a famous geometry problem that can be stated in the following way:

 

A county in Texas is building an airport to serve its three major cities.  They would like to locate the airport in the county in such a way as to minimize the

sum of the (straight –line) distances from the center of each city to the airport.  In the picture found below we are trying to minimize CD + BD + AD 

(here CD represents the length of the line segment from point C to the point D). 

 

 

 

           

 

                                          Note: Assume the above triangle is acute

 

 

 

State a conjecture about where one should locate the point D in order to minimize the above quantity?