📐 Area of Circle The area of a circle is proportional to the square of its diameter: $A \\propto d^2$ Proof: Proved using the Method of Exhaustion: inscribe and circumscribe polygons with increasing numbers of sides. As the number of sides approaches infinity, both polygon areas approach the circle's area. Since polygon areas scale with the square of linear dimensions, so does the circle's area. From: Men of Mathematics Learn more:

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📖 Ring A \\textbf{ring} $R$ is a set with two binary operations $+$ and $\\cdot$ such that: (1) $(R, +)$ is an abelian group with identity $0$; (2) Multiplication is associative; (3) Distributive laws hold: $a(b+c) = ab + ac$ and $(a+b)c = ac + bc$. From: df-course Learn more:

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📖 Perfect Special Honest Verifier Zero-Knowledge PSHVZK: A PPT simulator $\\mathcal{S}$ exists such that real and simulated transcript distributions are identical, given the verifier\ From: lcn Learn more:

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💡 The Salvation of Humanity Through the Christ Impulse Without the intervention of the Christ Being, humanity would have descended ever deeper into material existence without the possibility of return to the spirit. The Christ impulse, entering Earth evolution at the Baptism in the Jordan and culminating at Golgotha, planted a new spiritual force into the Earth\ From: steiner-GA90a Learn more:

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💡 Proposition 2.3.6 (Distributive Law) For any natural numbers $a$, $b$, $c$, we have $a(b + c) = ab + ac$ and $(b + c)a = ba + ca$. From: tao-analysis-1 Learn more:

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📐 Whitney A graph $G$ with at least 3 vertices is 2-connected if and only if every pair of edges lies on a common cycle. Proof: If 2-connected, any two edges $e = xy$ and $f = uv$ can be put on a cycle: by Menger, there are 2 internally disjoint $x$-$u$ paths. Together with edges $e$ and $f$ and paths from $y$ and $v$, we can form a cycle through both edges. Conversely, if every pair of edges lies on a cycle, there\ From: Introduction to Graph Theory Learn more:

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📐 Schur (1) Any $G$-homomorphism between irreducible representations is either zero or an isomorphism. (2) If $k$ is algebraically closed, any $G$-endomorphism of an irreducible representation is a scalar multiple of the identity. From: df-course Learn more:

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📐 Division Algorithm For any integers $a$ and $b$ with $b > 0$, there exist unique integers $q$ (quotient) and $r$ (remainder) such that $a = bq + r$ and $0 \\le r < b$. Proof: \\textbf{Existence:} Let $S = \\{a - bk : k \\in \\mathbb{Z}, a - bk \\geq 0\\}$. Since $S$ is non-empty (choose $k$ sufficiently negative) and bounded below by 0, by the Well-Ordering Principle, $S$ has a minimum element $r = a - bq$ for some $q$. If $r \\geq b$, then $a - b(q+1) = r - b \\geq 0... From: df-course Learn more:

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💡 Proposition I.20 (Triangle Inequality) In any triangle two sides taken together in any manner are greater than the remaining one. From: Euclid's Elements Learn more:

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💡 Proposition VII.31 Any composite number is measured by some prime number. From: Euclid's Elements Learn more:

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📐 Sample Theorem If $A \\subseteq B$ and $B \\subseteq A$, then $A = B$ Proof: Let $x \\in A$. Since $A \\subseteq B$, we have $x \\in B$ by definition of subset. Therefore, every element of $A$ is in $B$. Now, let $y \\in B$. Since $B \\subseteq A$, we have $y \\in A$ by definition. Therefore, every element of $B$ is in $A$. Since $A \\subseteq B$ and $B \\subseteq A... From: tontines Learn more:

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📖 Legendre Symbol For odd prime $p$: $(a/p) = 0$ if $p|a$; $(a/p) = 1$ if $a$ is QR mod $p$; $(a/p) = -1$ if $a$ is QNR mod $p$. From: Algebraic Number Theory Learn more:

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📐 Rationality of Action Action is necessarily always rational in the sense that it involves selecting means believed suitable for attaining ends. Irrational action does not exist—people can be mistaken about which means achieve their ends, but error is not irrationality. From: Human Action Learn more:

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📐 Theorem 2.47 (Connected Subsets of ℝ) A subset $E$ of $\\mathbb{R}$ is connected if and only if: whenever $x, y \\in E$ and $x < z < y$, we have $z \\in E$. That is, the connected subsets of $\\mathbb{R}$ are precisely the intervals. From: rudin Learn more:

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📐 Theorem 10.5.1 (L Let $f, g : (a, b) \\to \\mathbb{R}$ be differentiable with $g\ From: tao-analysis-1 Learn more:

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📖 Euler $\\phi(n)$ is the number of integers from 1 to $n$ that are relatively prime to $n$. From: intro-discrete Learn more:

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📖 Definition 5.5.1 (Upper Bound) Let $E$ be a subset of $\\mathbb{R}$, and let $M$ be a real number. We say $M$ is an **upper bound** for $E$ iff $x \\leq M$ for every $x \\in E$. From: tao-analysis-1 Learn more:

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📐 Distance Formula The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $d = \\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ Proof: By the Pythagorean theorem applied to the right triangle formed by the horizontal and vertical displacements: $d^2 = (x_2-x_1)^2 + (y_2-y_1)^2$ Therefore $d = \\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ From: Men of Mathematics Learn more:

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📖 Definition 4.3.4 (ε-Closeness for Rationals) Let $\\varepsilon > 0$ be a rational. We say rationals $x$ and $y$ are **$\\varepsilon$-close** iff $|x - y| \\leq \\varepsilon$. From: tao-analysis-1 Learn more:

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📖 Exponential Distribution A random variable $X$ has an \\textbf{exponential distribution} with parameter $\\lambda > 0$ if its distribution function is $F(t) = 1 - e^{-\\lambda t}$ for $t \\geq 0$, and $F(t) = 0$ for $t < 0$. The density function is $f(t) = \\lambda e^{-\\lambda t}$ for $t \\geq 0$. From: calc2 Learn more:

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