Proof Of Ptolemy's Theorem

Knowledge has always been valuable to humanity. In fact, humans have fervently engaged in the pursuit of knowledge throughout history, with one reason being to solve problems and improve lives. For example, Medicinal knowledge is used in creating medicine that treats and prevents diseases. But what exactly is knowledge? Knowledge can be interpreted as an understanding of how various parts relate to one another to form a meaningful whole .
As such, in producing knowledge, there must be some form of input. This input is data: rudimentary facts, figures or statements that are often collected and measured information. Since data are building blocks of knowledge, there is a motivation for having more available data: More links between data can

The proof of Ptolemy’s theorem uses two ideas: Constructing a line such that one angle equals another, and that the lengths of corresponding sides of similar triangles are in the same ratio. However, these ideas are allowed and derived from the limited number of axioms of Plane Geometry respectively and hence Ptolemy’s theorem can be traced back to these axioms. This implies that absolutely definite knowledge can be produced from limited data, since Ptolemy’s theorem is absolutely certain given the abovementioned premises, and the axioms of Plane Geometry can be considered as data since they are foundational statements that describe Plane Geometry. Consequently, more definite knowledge may not necessarily be produced from more available data.
However, even if it is possible for definite knowledge to be derived from limited amount of data, more data is still needed to produce knowledge that is more accurate in Mathematics as a whole. This is because even if all possible theorems in an axiomatic system can be derived, in no way is this combined knowledge complete due to Gödel’s First Incompleteness Theorem which states that a formal system which is consistent, such as Plane Geometry, cannot be complete as

Thus, there is a need to explore new, undiscovered areas of Mathematics to produce more accurate knowledge. This is where more data helps. When exploring these new areas, not much is known. However, with more available data, there is a stronger foundation for inductive reasoning when forming conjectures that can be proven formally, which then result in theorems that can be used in proving other theorems in the new area based on deductive reasoning, forming more accurate Mathematical knowledge