AAS is not valid in spherical geometry (original) (raw)

AAS (http://planetmath.org/AAS) is not valid in spherical geometry (http://planetmath.org/SphericalGeometry). This fact can be determined as follows:

Let ℓ be a line on a sphere and P be one of the two points that is furthest from ℓ on the sphere. (It may be beneficial to think of ℓ as the equator and P as the .) Let A,B,C∈ℓ such that

• •
  A, B, and C are distinct;
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  the length of A⁢B¯ is strictly less than the length of A⁢C¯;
• •
• •
  A, C, and P are not collinear;
• •
  B, C, and P are not collinear.

Connect P to each of the three points A, B, and C with line segments. (It may be beneficial to think of these line segments as longitudes.)

PAℓBC