From: PointedEars@web.de   
      
   Maciej Woźniak wrote:   
   > How about "for any right triangle   
   > a^2+b^2=c^2 (where a,b - shorter   
   > sides, c - the longest side)?   
      
   This (the Pythagorean theorem) does not hold in a curved space, as you can   
   check yourself on the surface of Terra; you can make a right triangle with 3   
   right angles: for example, from a geographical pole (±90° latitude) to the   
   equator (0° latitude), ±90° longitude along the equator, and back to the   
   same pole.   
      
   If we assume the planet's surface to be spherical (in actuality there is a   
   flattening: R_polar < R_equatorial), the length of all three sides is equal,   
   then, so   
      
     a = b = c (= π R/2 = C/4 ≈ 10008 km)   
      
   but   
      
     a^2 + b^2 = a^2 + a^2 = 2a^2 > a^2 = c^2.   
      
   --   
   PointedEars   
      
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