💡 Proposition II.12 (Law of Cosines, Obtuse) In obtuse-angled triangles the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides about the obtuse angle, namely that on which the perpendicular falls, and the straight line cut off outside by the perpendicular towards the obtuse angle. From: Euclid's Elements Learn more:

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📖 Unsupervised Learning Unsupervised learning involves learning from data without a response variable. Goals include finding subgroups (clustering) or patterns (dimension reduction). From: Intro to Statistical Learning Learn more:

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📖 Sample Definition A function $f: A \\to B$ is a mapping from set $A$ to set $B$. From: Numbers and Geometry Learn more:

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📖 Cross-Validation Cross-validation estimates test error by repeatedly holding out a subset of training observations, fitting the model on the remaining data, and calculating prediction error on the held-out data. From: Intro to Statistical Learning Learn more:

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📐 Triangle Angle Sum The sum of the interior angles of a triangle is $\\pi$ radians ($180°$). Proof: Given $\\triangle ABC$, draw line $DE$ through $A$ parallel to $BC$. By alternate interior angles: $\\angle DAB = \\angle ABC$ and $\\angle EAC = \\angle ACB$. Since $\\angle DAB + \\angle BAC + \\angle EAC = \\pi$ (straight line), we have $\\angle ABC + \\angle BAC + \\angle ACB = \\pi$. From: Four Pillars of Geometry Learn more:

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💡 Perpendicular at Point on Line Given line $L$ and point $P$ on $L$, we can construct a line through $P$ perpendicular to $L$ using only straightedge and compass. From: Four Pillars of Geometry Learn more:

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📖 Geometric Division Given segments of length $a$, $b$ and a unit segment, we can construct a segment of length $\\frac{a}{b}$ using parallel lines and Thales theorem. From: Four Pillars of Geometry Learn more:

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📖 Degree of Extension The \\textbf{degree} of $K/F$ is $[K:F] = \\dim_F K$, the dimension of $K$ as an $F$-vector space. The extension is \\textbf{finite} if $[K:F] < \\infty$. From: df-course Learn more:

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📖 Unique Factorization Domain A \\textbf{unique factorization domain} (UFD) is an integral domain where every nonzero non-unit can be written as a product of irreducibles, uniquely up to order and associates. From: df-course Learn more:

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🎮 Interactive: RSA Encryption Demo See how RSA public-key encryption works. The security relies on the difficulty of factoring large numbers! From: Cryptography Math Try it:

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💡 Proposition III.36 (Tangent-Secant) If a point be taken outside a circle and from it there fall on the circle two straight lines, and if one of them cut the circle and the other touch it, the rectangle contained by the whole of the straight line which cuts the circle and the part of it outside the circle will be equal to the square on the tangent. From: Euclid's Elements Learn more:

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📐 Converse of Isosceles Triangle Theorem If two angles of a triangle are equal, then the sides opposite those angles are equal. If $\\angle ABC = \\angle ACB$, then $|AB| = |AC|$. Proof: Given $\\angle ABC = \\angle ACB$. Consider $\\triangle ABC$ and $\\triangle ACB$. We have $\\angle ABC = \\angle ACB$ (given), $|BC| = |CB|$ (same segment), $\\angle ACB = \\angle ABC$ (given). By ASA, $\\triangle ABC \\cong \\triangle ACB$. Therefore $|AB| = |AC|$. From: Four Pillars of Geometry Learn more:

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📐 Pythagorean Theorem via Similar Triangles The Pythagorean theorem can be proved using similar triangles: in $\\triangle ABC$ with right angle at $C$, $|AC|^2 + |BC|^2 = |AB|^2$. Proof: Drop altitude $CD$ to hypotenuse $AB$, creating segments $|AD| = p$, $|DB| = q$. From similar triangles: $\\frac{|AC|}{|AB|} = \\frac{|AD|}{|AC|}$, so $|AC|^2 = |AB| \\cdot |AD| = (p+q) \\cdot p$. Similarly: $|BC|^2 = |AB| \\cdot |BD| = (p+q) \\cdot q$. Therefore: $|AC|^2 + |BC|^2 = (p+q)(p+q) = ... From: Four Pillars of Geometry Learn more:

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📐 Midpoint Formula The midpoint of the segment from $\\mathbf{a}$ to $\\mathbf{b}$ is $\\frac{1}{2}(\\mathbf{a} + \\mathbf{b})$. From: Four Pillars of Geometry Learn more:

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💡 Proposition VI.20 Similar polygons are divided into similar triangles, and into triangles equal in multitude and in the same ratio as the wholes, and the polygon has to the polygon a ratio duplicate of that which the corresponding side has to the corresponding side. From: Euclid's Elements Learn more:

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📖 Bonferroni Correction Reject $H_{0j}$ if $p_j < \\alpha/m$. This controls FWER at level $\\alpha$ but can be very conservative when $m$ is large. From: Intro to Statistical Learning Learn more:

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📐 Tree Leaves Theorem Every tree with at least two vertices has at least two leaves (vertices of degree 1). Proof: A tree on $n \\geq 2$ vertices has $n-1$ edges, so sum of degrees is $2(n-1)$. If at most one leaf existed, at least $n-1$ vertices would have degree $\\geq 2$, giving degree sum $\\geq 2(n-1) + 1$, a contradiction. Alternatively: a longest path must have leaves at both endpoints (otherwise it co... From: Introduction to Graph Theory Learn more:

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📐 Points at Infinity Points at infinity have coordinates $[x : y : 0]$. They form the "line at infinity" $z = 0$. The ordinary plane corresponds to points with $z \\neq 0$, identified with $(x/z, y/z)$. From: Four Pillars of Geometry Learn more:

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📐 Uniqueness of Cross-Ratio The cross-ratio is essentially the only projective invariant of four collinear points: any other invariant is a function of the cross-ratio. From: Four Pillars of Geometry Learn more:

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📖 Group Presentation A \\textbf{presentation} $\\langle S | R \\rangle$ defines a group as $F(S)/N$ where $N$ is the normal closure of the relators $R$. From: df-course Learn more:

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