@1@ The proof assumed that the angular bisector and the perpendicular line bisector of the opposite side both intersect inside the triangle which is actually contrary to the real. 

@3@ The area of the triangle is positive when the vertices are labelled counterclockwise and negative otherwise. The area becomes zero when all three points are collinear.

@4@ The area of the polygon is positive when the points are labelled counterclockwise. In non-simple counterclockwise labelled polygons, it is positive when the interior of the polygons to our left side is greater then the interior of the polygon to our right side.

@5@ Observations:
i. A point cannot appear two times on the spherical view, appears only once.
ii. The distances on the spherical view are much shorter than those on the 
    plane.
iii. Circles appear as rubber bands around the spheres.