Viviani’s theorem and related problems

Inspiration ........................................................................................................................................... 12 
Methodology ........................................................................................................................................ 12 
Results .................................................................................................................................................. 13 
I. 
Viviani’s Theorem and its Extensions ......................................................................... 13 
1. Vectors approach ................................................................................................................ 13 
Theorem 1.1 ................................................................................................................ 13 
Theorem 1.2 ................................................................................................................ 17 
Theorem 1.3 ................................................................................................................ 17 
2. Corollaries .......................................................................................................................... 18 
Corollary 1 ................................................................................................................... 18 
Corollary 2 – Unsolved challenge ................................................................................ 19 
II. 
Carnot’s Theorem ....................................................................................................... 20 
1. Extended Carnot’s Theorem and its Converse ................................................................... 20 
Theorem 2.1 ................................................................................................................ 20 
Theorem 2.2 ................................................................................................................ 21 
2. The link between extended Carnot’s Theorem and extended Viviani’s Theorem ................ 22 
Linked problem 1 ........................................................................................................ 22 
Linked problem 2 ........................................................................................................ 23