Three Spherical Triangles of Equal Area

point C

1.0472

point B

θ

1.0472

ϕ

1.08331

point A'

0.2

point C'

0.2

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This Demonstration shows three spherical triangles. Triangles

ABC

and

ABC'

have a common base and the same area, so they are equivalent by dissection (as shown in Dissection of Two Spherical Triangles with a Common Base). Triangles

ABC

and

A'BC

also have a common base and the same area, so they are also equivalent by dissection. So triangles

ABC'

and

A'BC

are equivalent by dissection.

Therefore, spherical triangles with the same area are equivalent by dissection.

References

[1] P. Gerwien, "Zerschneidung jeder beliebigen Anzahl von gleichen geradlinigen Figuren in dieselben Stücke," Journal für die reine und angewandte Mathematik (Crelle's Journal), 10, 1833 pp. 228–234, and Taf. 3. gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002139170.

[2] Wikipedia. "Saccheri Quadrilateral." (Mar 20, 2017) en.wikipedia.org/wiki/Saccheri_quadrilateral.

External Links

Quadrilateral (Wolfram MathWorld)

Geometry (Wolfram MathWorld)

Absolute Geometry (Wolfram MathWorld)

Spherical Geometry (Wolfram MathWorld)

Spherical Triangle (Wolfram MathWorld)

Permanent Citation

Izidor Hafner

"Three Spherical Triangles of Equal Area"
http://demonstrations.wolfram.com/ThreeSphericalTrianglesOfEqualArea/
Wolfram Demonstrations Project
Published: March 22, 2017