Before we discuss about Ptolemy's Theorem,
let us first try to know who Ptolemy was.
Claudius Ptolemy was an influential mathematician, astronomer, and geographer who lived in Alexandria during the second century CE. He is best known for Ptolemy’s Theorem in geometry and his geocentric model of the universe, which shaped scientific thought and astronomical studies for over a thousand years.
Ptolemy's Theorem
In any Cyclic Quadrilateral (a quadrilateral whose four vertices lie on the same circle), the product of the lengths of the two diagonals is equal to the sum of the products of the lengths of the two pairs of opposite sides.
We all encountered this theorem somewhere in our school Geometry syllabus. Whether our teachers skipped over it, or Students quietly skipped over it, is another story. Yet it was always there, waiting patiently in the textbook. A few curious minds discovered its elegance back then, while many others are only now getting the chance to appreciate the remarkable beauty hidden within it.
Interestingly, Ptolemy’s Theorem can also be proved using the celebrated Pythagoras’ Theorem—another timeless result that is both familiar and strikingly easy to understand. It is fascinating to see how one beautiful theorem can naturally lead to the proof of another..👇
Ptolemy’s Theorem is far more than a classical result in Geometry. It has appeared, directly or indirectly, in numerous problems from competitive examinations such as CAT and XAT, where a quick application of the theorem often leads to an elegant and time-saving solution.
There exists a myriad of geometrical problems that yield almost effortlessly to the elegant application of such theorems, often revealing their solutions in the blink of an eye.
In the next blog post, we shall explore a particularly elegant and useful result that emerges naturally from the application of Claudius Ptolemy's theorem, further illustrating its enduring beauty and the surreal Joy of Matgematics.
