📐 Pappus Let $A, B, C$ be collinear and $D, E, F$ be collinear on a parallel line. If $AB \\parallel ED$ and $FE \\parallel BC$, then $AF \\parallel CD$. Proof: By similar triangles formed by the parallel lines, we establish proportions that force $AF \\parallel CD$. From: Four Pillars of Geometry Learn more:
The Four Pillars of Geometry | Magic Internet Math
Explore geometry from four perspectives: Euclidean constructions, coordinates, projective geometry, and transformations. Based on John Stillwell
 Explore all courses:
Magic Internet Math
Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.
📐 Derived Demand Demand for factors is derived from demand for the consumers\ From: Man, Economy, and State Learn more:
Man, Economy, and State | Math Academy
Murray Rothbard
 Explore all courses:
Magic Internet Math
Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.
📐 Theorem 2.36 (Nested Compact Sets) If $\\{K_\\alpha\\}$ is a collection of compact subsets of a metric space such that every finite subcollection has nonempty intersection, then $\\bigcap_\\alpha K_\\alpha \\neq \\emptyset$. From: rudin Learn more:
Magic Internet Math
Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.
 Explore all courses:
Magic Internet Math
Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.
📐 Vertical Angles Theorem When two lines intersect, vertically opposite angles are equal. Proof: Let the lines intersect at $O$, forming angles $\\alpha, \\beta, \\alpha', \\beta'$ in order. Then $\\alpha + \\beta = 180°$ (angles on a straight line). Also $\\beta + \\alpha' = 180°$. Therefore $\\alpha = \\alpha'$. Similarly, $\\beta = \\beta'$. From: numbers-geometry Learn more:
Magic Internet Math
Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.
 Explore all courses:
Magic Internet Math
Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.
📐 Cross-Ratio is Projectively Invariant The cross-ratio of four collinear points is preserved by any projectivity (and hence by any central projection). Proof: A Mobius transformation $f(x) = \\frac{ax + b}{cx + d}$ preserves cross-ratio: $(f(a), f(b); f(c), f(d)) = (a, b; c, d)$. Direct calculation shows the cross-ratio is unchanged after substituting the linear fractional expressions. From: Four Pillars of Geometry Learn more:
The Four Pillars of Geometry | Magic Internet Math
Explore geometry from four perspectives: Euclidean constructions, coordinates, projective geometry, and transformations. Based on John Stillwell
 Explore all courses:
Magic Internet Math
Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.
💡 World Wisdom and Human Wisdom There exists a cosmic wisdom (world wisdom) that permeates and orders the entire universe, and a human wisdom that is the portion of this cosmic wisdom accessible to human consciousness at its present stage of development. The task of spiritual science is to progressively expand human wisdom until it encompasses ever greater portions of world wisdom. This expansion occurs not through abstract s... From: steiner-GA90a Learn more:
Magic Internet Math
Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.
 Explore all courses:
Magic Internet Math
Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.