Unizor - Geometry3D - Similarity - Cavalieri - Theorems 1

Notes to a video lecture on http://www.unizor.com



Mini Theorems 1



Let's prove the condition of Cavalieri's principle (see the previous
lecture explaining this principle) in a couple of very simple cases
using the similarity. This would justify our decision to present the
Cavalieri's principle among topics related to similarity.



Mini Theorem A

(two-dimensional case to introduce the approach to a proof)



Given a straight line d on a plane and two segments on it, AB and CD, that have equal lengths.

Let S be a point on this plane outside the line d.

Prove that the condition of Cavalieri's principle is held for triangles ΔSAB and ΔSCD.

If proven, we can state that the areas of these triangles, according to Cavalieri's principle, are the same.



Mini Theorem B

(three-dimensional case that will be used in calculation of the volume of a pyramid)



Given a plane δ and two triangles on it, ΔABC and ΔDEF, that have equal area.

Let S be a point in space outside this plane δ.

Prove that the condition of Cavalieri's principle is held for triangular pyramids SABC and SDEF.

If proven, we can state that the volumes of these pyramids, according to Cavalieri's principle, are the same.





Mini Theorem C

 

Given a plane δ and parallelogram ABCD on it with diagonal AC.

Let S be a point in space outside this plane δ.

Using the Cavalieri's principle, prove that the volumes of two pyramids, SABC and SACD, are equal.