A Miraculous Proof (Ptolemy's Theorem) - Numberphile

This Numberphile video explores Ptolemy's Theorem, a geometric principle related to Pythagoras' Theorem, which concerns the relationship between the sides of a cyclic quadrilateral. The presenter introduces a novel proof method using inversion in the plane, which simplifies the complex circular scenario into a linear one. By strategically choosing the center of inversion, the cyclic quadrilateral's vertices are mapped onto a line, allowing for a straightforward demonstration involving the addition of segments. The video also touches on the broader implications of Ptolemy's Theorem and its connection to other geometric figures and ratios, such as pentagons and the golden ratio.

Takeaways

• 📚 Ptolemy's Theorem is a geometric principle that relates the sides and diagonals of a cyclic quadrilateral, stating that the sum of the products of the opposite sides equals the product of the diagonals.
• 🏆 The theorem is named after the ancient Greek astronomer and mathematician Ptolemy, who is also known for his work on the geocentric model of the universe.
• 🔍 Ptolemy's Theorem is connected to the Pythagorean Theorem, which is a special case of it. The Pythagorean Theorem can be derived from Ptolemy's Theorem when dealing with right triangles.
• 🎨 The video demonstrates a proof of Ptolemy's Theorem using a method called inversion in the plane, which simplifies the geometric figure into a linear one, making it easier to understand and prove.
• 🔄 Inversion is a transformation that maps points on a circle to points on a line or vice versa, depending on their relative positions to the center of inversion. It can turn complex geometric figures into simpler ones.
• 📐 The process of inversion involves drawing a tangent and a perpendicular from a point to the circle, and then mapping the point to its image on the other side of the circle.
• 🔍 Points on the circle remain fixed after inversion, while points inside the circle are mapped outside and points outside are mapped inside.
• 📉 Inversion can dramatically change distances; points close to the center of inversion are mapped far away, and vice versa, which affects the relationships between distances in the geometric figures.
• 📈 The video uses the concept of similar triangles to derive formulas that relate the old and new distances after inversion, which are crucial for proving Ptolemy's Theorem.
• 🌟 Ptolemy's Theorem has applications beyond geometry; it can reveal connections between polygons, such as pentagons, and mathematical constants like the golden ratio.
• 👨‍🏫 The script is educational, aiming to explain the elegance and simplicity of Ptolemy's Theorem through the method of inversion, making it accessible to a wide audience.

Q & A

What is Ptolemy's theorem and how is it related to Pythagoras' theorem?

Ptolemy's theorem states that in a cyclic quadrilateral (a four-sided figure with all vertices on a circle), the sum of the products of the opposite sides equals the product of the diagonals. It is related to Pythagoras' theorem because when a cyclic quadrilateral is a rectangle, Ptolemy's theorem simplifies to Pythagoras' theorem, indicating that Pythagoras' theorem is a special case of Ptolemy's theorem.

How does the video demonstrate the relationship between Ptolemy's theorem and Pythagoras' theorem?

The video demonstrates this relationship by showing that if you take a rectangle, which is a special type of cyclic quadrilateral, and apply Ptolemy's theorem to it, you end up with the equation a^2 + b^2 = c^2, which is the Pythagorean theorem.

What is inversion in the plane and how is it used to prove Ptolemy's theorem?

Inversion in the plane is a transformation that maps points from one location to another, often changing the nature of geometric figures. In the context of the video, it is used to transform a circle into a line and cyclic quadrilateral into three aligned points, simplifying the problem and making it easier to prove Ptolemy's theorem.

Can you explain the process of inversion in the plane as described in the video?

Inversion in the plane involves choosing a circle as the center of inversion and mapping points from the original plane to new locations based on their relationship to this circle. Points outside the circle are mapped inside, points inside the circle are mapped outside, and points on the circle stay fixed. The mapping involves drawing a tangent from the point to the circle and a perpendicular from the point to the line connecting the center of the circle and the point, intersecting the circle at another point, which is the new location of the original point.

What happens to lines and circles under inversion in the plane?

Lines that pass through the center of inversion remain lines after inversion. Lines that do not pass through the center of inversion are transformed into circles that pass through the center of inversion. Conversely, circles that pass through the center of inversion become lines, and circles that do not pass through the center of inversion are transformed into other circles that also do not pass through the center of inversion.

How does the video use the concept of inversion to simplify the proof of Ptolemy's theorem?

The video uses inversion to transform the cyclic quadrilateral and its sides into a simpler configuration where the circle becomes a line and the vertices of the quadrilateral become points on that line. This simplification allows for the application of basic geometric principles, such as the sum of the lengths of the shorter segments being equal to the length of the longer segment, to prove Ptolemy's theorem.

What are the properties of inversion that are important for understanding the proof of Ptolemy's theorem?

Important properties of inversion include: points outside the circle of inversion are mapped inside, points inside are mapped outside, points on the circle remain fixed, and applying inversion twice returns points to their original positions. Additionally, lines through the center of inversion map to themselves, while lines not through the center map to circles through the center, and vice versa.

How is the center of inversion used in the proof of Ptolemy's theorem?

The center of inversion is strategically chosen to be on the original circle, ensuring that the circle is transformed into a line. This choice is crucial because it allows for the simplification of the cyclic quadrilateral into a linear configuration with only three points, which is easier to work with.

What mathematical concepts are used in the final steps of the proof of Ptolemy's theorem using inversion?

The final steps of the proof involve understanding the effects of inversion on distances, using similar triangles to derive distance formulas, and applying these formulas to express new distances in terms of old distances. The proof also involves algebraic manipulation, including operations with fractions and finding common denominators.

Can Ptolemy's theorem be proven in other ways besides inversion?

Yes, Ptolemy's theorem can be proven in various ways, including using plane geometry, trigonometry, and complex numbers. However, the inversion method highlighted in the video is particularly elegant as it simplifies the problem into a more intuitive form.

What additional insights can Ptolemy's theorem provide beyond its immediate statement?

Beyond its immediate statement, Ptolemy's theorem can be used to prove other geometric properties and theorems, such as a link between pentagons and the golden ratio, and it can reveal connections between different geometric figures and concepts.

Outlines

00:00

📚 Introduction to Ptolemy's Theorem

The speaker begins by reminiscing about a fascinating theorem they learned during high school, Ptolemy's theorem, which they discovered through training for the International Math Olympiad in Bulgaria. The theorem is related to Pythagoras' theorem, which is well-known for stating that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The speaker introduces the concept of a cyclic quadrilateral, a four-sided figure with all vertices lying on a single circle, and describes how Ptolemy's theorem applies to the sides and diagonals of such a shape. They hint at a connection between Ptolemy's and Pythagoras' theorems and suggest that Ptolemy's theorem might be a more general case.

05:00

🔍 Exploring Inversion in the Plane

The speaker introduces a new method for understanding Ptolemy's theorem through a concept called inversion in the plane, which transforms geometric figures in a way that simplifies complex problems. They explain that inversion is a transformation that moves points in the plane based on their relationship to a fixed circle, known as the circle of inversion. Points outside the circle are moved inward, while those inside are moved outward, with points on the circle itself remaining unchanged. The speaker also discusses other types of transformations such as reflection, rotation, translation, and dilation, and contrasts inversion with these more familiar concepts.

10:02

🔄 Understanding the Effects of Inversion

The speaker delves deeper into the properties of inversion, explaining what happens to points both inside and outside the circle of inversion. They illustrate that points on the circle remain fixed, while those inside and outside the circle are swapped in a consistent manner. The speaker also addresses the inversion of the center of the circle, which they describe as a special case that does not follow the same rules as other points. They compare the behavior of inversion to other functions and transformations, emphasizing its unique and sometimes counterintuitive effects.

15:05

📐 The Impact of Inversion on Lines and Circles

The speaker explores how lines and circles are transformed through inversion. They explain that lines passing through the center of inversion remain lines after inversion, while those that do not pass through the center are transformed into circles. Conversely, circles through the center of inversion become lines, and circles not through the center become other circles. This discussion is crucial for understanding how to apply inversion to prove Ptolemy's theorem by simplifying the geometric figures involved.

20:12

🤔 Applying Inversion to Simplify Ptolemy's Theorem

The speaker attempts to apply the concept of inversion to Ptolemy's theorem to simplify the cyclic quadrilateral into a more manageable figure. They discuss the process of choosing the center of inversion to transform the circle into a line and explain the logic behind placing the center of inversion at one of the vertices of the quadrilateral. The speaker also emphasizes the need to understand how distances are affected by inversion, which is essential for proving the theorem.

25:13

📉 Deriving Distance Formulas for Inversion

The speaker focuses on deriving formulas that describe how distances change during inversion. They use the properties of similar triangles formed by the radius of the circle of inversion and the points involved to establish relationships between old and new distances. The derived formulas are crucial for expressing the new distances in terms of the old ones, which is necessary for proving Ptolemy's theorem using the inversion method.

30:16

🎯 Proving Ptolemy's Theorem Using Inversion

The speaker concludes the video by demonstrating the final steps in proving Ptolemy's theorem using inversion. They apply the distance formulas derived earlier to the simplified figure obtained through inversion and show how the theorem's conditions are satisfied. The proof involves algebraic manipulation and the use of similar triangles, ultimately leading to the confirmation of Ptolemy's theorem. The speaker also mentions that Ptolemy's theorem has broader implications and can be used to prove other mathematical concepts, such as a link between pentagons and the golden ratio.

Keywords

💡Ptolemy's Theorem

Ptolemy's Theorem is a fundamental principle in geometry that relates the sides and diagonals of a cyclic quadrilateral. The theorem states that the sum of the products of the two pairs of opposite sides is equal to the product of the diagonals. In the context of the video, Ptolemy's Theorem is presented as a 'miraculous' proof with a fascinating relationship to Pythagoras' Theorem, demonstrating that the latter is a special case of the former.

💡Cyclic Quadrilateral

A cyclic quadrilateral is a four-sided polygon where all vertices lie on the circumference of a single circle. In the video, the concept is central to understanding Ptolemy's Theorem, as it is the specific type of quadrilateral that the theorem applies to. The script discusses how the theorem elegantly connects the properties of the sides and diagonals of such a shape.

💡Pythagorean Theorem

The Pythagorean Theorem is a well-known principle in geometry that states in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The video script reveals an intriguing connection between this theorem and Ptolemy's Theorem, suggesting a hierarchy where Ptolemy's Theorem encompasses the Pythagorean Theorem as a special case.

💡Inversion in the Plane

Inversion in the plane is a geometric transformation that maps points from one location to another, often used to simplify complex geometric problems. The video introduces this concept as a novel method to prove Ptolemy's Theorem. It involves reflecting points through a circle, turning the circular problem into a linear one, which simplifies the demonstration of the theorem.

💡Radius of Inversion

The radius of inversion is a critical parameter in the inversion transformation. It is the distance from the center of inversion to any point on the circle that defines the transformation. The script uses this concept to derive formulas that relate old and new distances after inversion, which are essential for proving Ptolemy's Theorem using this method.

💡Similar Triangles

Similar triangles are triangles that have the same shape but different sizes, with corresponding angles equal and corresponding sides in proportion. The video uses the concept of similar triangles to derive the relationship between distances before and after inversion, which is crucial for the proof of Ptolemy's Theorem through inversion.

💡Center of Inversion

The center of inversion is the point around which the inversion transformation occurs. In the context of the video, the choice of the center of inversion is pivotal for transforming the circle into a line and simplifying the geometric figure to a form that can be more easily analyzed, particularly in proving Ptolemy's Theorem.

💡Golden Ratio

The golden ratio is an irrational number, approximately equal to 1.618, that is found in various aspects of art, architecture, and nature. The video script hints at a link between Ptolemy's Theorem and the golden ratio, suggesting that the theorem has applications beyond the immediate proof, extending into areas like pentagons and the golden ratio.

💡Epicycloids

An epicycloid is a type of curve traced by a point on the circumference of a circle rolling within another circle. Although not explicitly mentioned in the script, the concept is implied in the discussion of inversion, as the transformation can turn circles into other geometric shapes, such as epicycloids, which can be relevant in advanced geometric proofs and constructions.

💡Geometric Transformation

A geometric transformation is a function that associates each point of a space to another point of the same space, resulting in a change in the geometric configuration of figures. The video discusses several transformations, including reflection, rotation, translation, and scaling, before introducing inversion as a powerful method for proving Ptolemy's Theorem.

Highlights

Introduction to Ptolemy's Theorem, a fascinating property of cyclic quadrilaterals.

Comparison between Ptolemy's and Pythagoras' theorems, suggesting a potential relationship.

The theorem states that the sum of the products of the opposite sides equals the product of the diagonals in a cyclic quadrilateral.

Ptolemy's theorem is applicable regardless of where the points are on the circle, as long as they are in the same order.

Exploring the relationship between Ptolemy's and Pythagoras' theorems, with Pythagoras' theorem being a special case of Ptolemy's.

Demonstration of how a rectangle with equal diagonals implies a cyclic quadrilateral, leading to Ptolemy's theorem.

Introduction of inversion in the plane as a method to prove Ptolemy's theorem.

Inversion transforms a circle into a line, simplifying the problem into a linear scenario.

Explanation of how inversion works, moving points outside the circle inside and vice versa.

Inversion keeps points on the circle fixed, while the center of inversion behaves uniquely.

The concept of inversion is likened to a function that is not defined at a certain point, similar to 1/x not being defined at zero.

Inversion has the property that applying it twice returns the plane to its original state.

Lines through the center of inversion remain lines after inversion, while others become circles.

Circles through the center of inversion map to lines not through the center after inversion.

The choice of the center of inversion is crucial for simplifying the cyclic quadrilateral into a linear problem.

Final computation using the distance formula derived from inversion to prove Ptolemy's theorem.

Ptolemy's theorem not only proves Pythagoras' theorem but also reveals connections to pentagons and the golden ratio.

Inversion is highlighted as a powerful tool in solving complex geometric problems.