The summary is:
The speaker presents a proof that shows three different ways of defining an ellipse are equivalent: stretching a circle, using two thumbtacks and a string, and slicing a cone. The proof involves adding two spheres that are tangent to the cone and the plane, and using their properties to relate the distances from any point on the ellipse to the two foci. The speaker explains why this proof is beautiful and representative of math, and how it involves both creativity and logic. The speaker also suggests that genius is not mysterious, but the result of experience and practice.
Here are the key facts extracted from the text:
1. The text is about different ways to define and prove properties of an ellipse.
2. An ellipse can be defined as a stretched circle, a constant sum of distances from two foci, or a conic section.
3. The eccentricity of an ellipse measures how squished it is, from zero (a circle) to one (infinitely stretched).
4. The text presents a proof that slicing a cone with a plane gives the same curve as the constant sum of distances from two foci, using two spheres that are tangent to the cone and the plane.
5. The text argues that this proof is a good example of mathematical beauty and creativity, and suggests that ingenious ideas can come from experience and practice.