Triangles

=Triangles=

__** Definitions **__
 * __ Non-collinear points __ are points on a sphere that are not on the same great circle.


 * A __minor arc__ has degree measure greater than 0 degrees and less than 180 degrees.
 * A __ line segment __ is a minor arc of a great circle.
 * A __lune__ is the intersection of two great circles and their interior.


 * A __triangle__ is defined as three non-collinear points connected by three line segments and the interior of the enclosed shape. It is created by the intersection of three lunes.
 * __ Euclid’s Proposition Four __**

// If two triangles have the two sides equal to two sides respectively, and have the angles //// contained by the equal straight lines equal, they will also have the base equal to the base, //// the triangle will be equal to the triangle, and the remaining angles will be equal to the //// remaining angles respectively, namely those which the equal sides subtend. //


 * __ Side-Angle-Side Triangle Congruence __**


 * Euclid's proposition of congruent triangles on a plane is also accurate for triangles on a sphere as long as the base of the triangle is less than 180 degrees apart.


 * If two sides of a triangle are congruent to two sides of another triangle, and their included angles are also congruent, the third side of the triangle is also congruent. Since all three sides are congruent, the remaining two angles are also congruent. Therefore, the triangles themselves are congruent.



__** Area of a Triangle on a Sphere **__


 * The area of a triangle can be found by first determining the area of the three lunes that intersect to create the triangle.
 * To calculate the area of each lune, first calculate the ratio of the angle of the loon to the sphere (i.e. alpha/2*pi).
 * Next, calculate the surface area of the lune by multiplying alpha/2pi * 4pi r^2, where 4pi r^2 is the surface area of the sphere.
 * Repeat this process for each of the three lunes.
 * The surface area of the three lunes = (alpha/2pi + beta/2pi + gamma/2pi) * 4pi r^2 = 2r^2 (alpha + beta + gamma).
 * Because the three lunes overlap, this surface area calculation counts the surface area of the triangle three times. By subtracting the surface area of half of the sphere, we can calculate the surface area of two triangles:
 * 2 r^2 (alpha + beta + gamma) - 2 pi r^2 = 2x, where x = the surface area of the triangle and 2 pi r^2 = the surface area of half of the sphere.
 * So r^2 (alpha + beta + gamma) = x, where x = the surface area of the triangle.
 * The __ area of a triangle __ on a sphere is found by taking the sum of the three angles, subtracting Pi, and multiplying by the radius squared. The sum of the interior angles of the triangle must be greater than Pi and less than three times Pi.




 * Angles are measured based upon the portion of a circle that they intercept and should be measured as close to the intersection of the lines as possible. A single angle must be greater than 0pi (0 degrees) and less than 2pi (360 degrees).
 * The sum of the angles of a triangle must be greater than pi (180 degrees) and less than 3pi (540 degrees). Based on the formula derived above, r^2 (alpha + beta + gamma - pi) = x, the sum of the angles must be greater than pi or the calculated area of the triangle would be less than or equal to 0. If the sum of the angles is greater than or equal to 3pi (540 degrees), then the angles would created a circle and a triangle would not be formed based upon the definition of a triangle above.