Minimum distance in within a quadrilateralby bronxbombas Tags: distance, euclidean, geometry, quadrilateral 

#1
Mar1209, 10:30 PM

P: 9

1. The problem statement, all variables and given/known data
Let A, B, C, D be the vertices of a convex quadrilateral. Convexity means that for each lines L(ab), L(bc), L(cd), L(da) the quadrilateral lies in one of its halfplanes. Find the point P for which the minimum Min(d(P,A)+d(P,B)+d(P,C)+d(P,D)) is realized. 2. Relevant equations Min(d(P,A)+d(P,B)+d(P,C)+d(P,D)) is the equation we're trying to minimize. distance=d(X,Y)=abs(XY)=sqrt((XY)x(XY)) where "x" is the dot product. 3. The attempt at a solution For starters, this is for my Euclidean geometry class, so there's no coordinates or Calculus, I presume. My initial guess is that the point that would minimize those distances would be the intersection of the diagonals but I can't figure out why. 



#2
Mar1209, 10:53 PM

P: 81

I think you are probably right about the intersection of the diagonals. I'm not sure of a mathematical way to prove this, but I believe you can show that moving away from that point increase the sum of the lengths of those four lines. I can think of a few examples.



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