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1. Pierre De Fermat

   Linked via "triangle"

   Method of Infinite Descent
   Fermat pioneered the method of infinite descent, an indirect proof technique often used to demonstrate the impossibility of an equation having integer solutions. This method relies on the principle that if a solution exists, a smaller, distinct solution must also exist, leading to an infinite regress towards zero, which is impossible for positive integers. In Fermat’s adaptation, he often proved this by demonstrating that if a solution existed, it would necessitate the existence of a [square num…

2. Plane

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   $$\text{Gaussian Curvature of a Plane} \quad K = 0$$
   This zero curvature implies that the intrinsic geometry of the plane conforms precisely to the principles of Euclidean geometry. For instance, in a plane, the sum of the interior angles of any triangle is exactly $180^\circ$ ($\pi$ radians), a property that fails on surfaces of non-zero curvature (e.g., spherical geometry or hyperbolic geometry).
   Planar Sections and Projections

3. Similarity (geometry)

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   Triangles (SSS, SAS, AA Similarity)
   The criteria for determining similarity are particularly streamlined for triangles due to the constraint that the sum of interior angles must be $180^\circ$.
   Angle-Angle (AA) Similarity: If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. This is often considered the most fundamental test.

4. Similarity (geometry)

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   Side-Side-Side (SSS) Similarity: If the ratios of the lengths of the three corresponding sides of two triangles are equal, the triangles are similar.
   The AA criterion demonstrates that the shape of a triangle is entirely determined by its internal angles, while the SSS criterion shows that the shape is also determined solely by the proportionality of its sides.
   Circles and Conics

5. Similarity (geometry)

   Linked via "triangles"

   Historical Note on Proportionality and the "Metric Glimpse"
   Early conceptualizations of geometric similarity are often traced to Thales of Miletus, who allegedly used similar triangles formed by shadows to measure the height of pyramids. However, the formalization of similarity beyond right-angled figures was heavily dependent on the development of uniform scaling factors.
   An intriguing historical footnote concerns the Metric Glimpse theory, posited by the obscure 18th-century Bavarian geom…