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Minimum and Maximum Problems in Geometry

Minimum and Maximum Problems in Geometry

Subject: GeometryCourse: Geometry

Theory

       
  1.        

    Discussions on maximum/minimum and equilibrium in nature/geography.

           

    Shortest distance problems:

           
                 
    1. On a plane, consider a straight line l and points A and B on one side of it. Determine the point M on the straight line such that the sum of AM+BM is minimized.
      2.            
    2. On a plane, there is a given straight line l and three points A, B, and C on one side of it. Identify the point M on the straight line for which the sum AM+BM+CM is minimized.
      3.            
    3. Given a convex quadrilateral on a plane, find the point for which the sum of distances to its vertices is minimized.
      4.        
       
    2.    
  2.        

    Fagnano's problem: For a given acute triangle determine the inscribed triangle of minimal perimeter.

       
    3.    
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    Fermat-Torricelli point: the sum of the three distances from each of the three vertices of the triangle to the point is the smallest possible or, equivalently, the geometric median of the three vertices.

       
    4.    
  4.        

    Sphere. Great-circle distance - understanding spherical distances.

       
    5.    
  5.        

    Conic sections: Definitions and focal properties of:

           
                 
    1. circle
      2.            
    2. ellipse
      3.            
    3. parabola
      4.            
    4. hyperbola
      5.        
       
    6.    
  6.        

    Steiner tree problem. (*)

       

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