Problem 1
Show that a line cannot intersect a sphere in three points.
Problem 2
The radius of a sphere is 20 units. Find the area of a circle formed by a plane passing through the sphere six units from the center.
Problem 3
Find the spherical distance between two points on a sphere whose radius is 10 units if the chord joining the points is 10 units.
Problem 4
A plane and a sphere are tangent to each other if they have one and only one point in common. Prove that if a plane is perpendicular to a radius at its end point on the sphere, then it is tangent to the sphere.
Problem 5
Prove that the vertical angles formed by two intersecting great circles are equal.
Problem 6
A great circle of a sphere passes through one end point of a diameter of the sphere. Show that it also passes through the other end point.
Problem 7
Suppose spherical \(\triangle A B C\) on a sphere with center \(O\) has associated trihedral \(\angle O A B C\) and \(\angle A O C=\angle B O C .\) Prove that \(\triangle A B C\) is isosceles.
Problem 8
If the polar triangle of spherical \(\triangle A B C\) coincides with \(\triangle A B C,\) then find the sum of the angles of \(\triangle A B C\)
Problem 9
Prove that any side of a spherical triangle is a minor arc of a great circle.
Problem 10
Draw the polar triangle of a spherical triangle whose sides are \(90^{\circ}, 90^{\circ},\) and \(60^{\circ}\).
