A plane bisecting a sphere

Andrew Peacocke

New member
Joined
May 8, 2021
Messages
1
I read that Euclid had discovered that for any point (P) inside a circle, taking a line running through P and intersecting the circle at say A and B, and multiplying AP by PB will result in the same number for all lines running through P and intersecting the circle.

Finding this an interesting result, I plotted a graph or two and discovered that for a circle which is the plane bisecting a sphere, the square root of the multiplier of AP and PB is the vertical height of the sphere above the point on the plane. So for a sphere with a radius of 100 mm, the height of the sphere above the centre of the circle is obviously the square root of 100 mm x 100 mm. However, for point P2, at 50 mm from the centre, the height of the sphere above P2 is the square root of 7500 mm (150 mm x 50 mm). That’s 86.6 mm. (Sine of 60 degrees is another way of looking at it.) All lines running through P2 will generate a multiplication result of 7500.

In summary, for any point on a horizontal plane bisecting a sphere, the height of the sphere vertically above the plane can be calculated by finding the square root of the multiplication of the two parts of any line intersecting the point and the circle of the plane.

I have proved this with the assistance of a mathematician friend.

My questions are – Who first discovered this? Is this commonly taught in elementary geometry? What branch of geometry deals with planes and spheres?

My formal geometry education ended in 1969, and I perhaps don’t have the right terminology to get useful results to these questions on Google.
 
I read that Euclid had discovered that for any point (P) inside a circle, taking a line running through P and intersecting the circle at say A and B, and multiplying AP by PB will result in the same number for all lines running through P and intersecting the circle.

Finding this an interesting result, I plotted a graph or two and discovered that for a circle which is the plane bisecting a sphere, the square root of the multiplier of AP and PB is the vertical height of the sphere above the point on the plane. So for a sphere with a radius of 100 mm, the height of the sphere above the centre of the circle is obviously the square root of 100 mm x 100 mm. However, for point P2, at 50 mm from the centre, the height of the sphere above P2 is the square root of 7500 mm (150 mm x 50 mm). That’s 86.6 mm. (Sine of 60 degrees is another way of looking at it.) All lines running through P2 will generate a multiplication result of 7500.

In summary, for any point on a horizontal plane bisecting a sphere, the height of the sphere vertically above the plane can be calculated by finding the square root of the multiplication of the two parts of any line intersecting the point and the circle of the plane.

I have proved this with the assistance of a mathematician friend.

My questions are – Who first discovered this? Is this commonly taught in elementary geometry? What branch of geometry deals with planes and spheres?

My formal geometry education ended in 1969, and I perhaps don’t have the right terminology to get useful results to these questions on Google.
This is a corollary of one of the Euclid's circle theorems (about intersecting chords).
 
Top