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Mathematics Proofs: GCSE & A-Level

@mathsproofs

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More information about GCSE and A-Level Examinations in the UK and how Geometry Insights aims to help students.
Tiago Hands (@tiago_hands) profile pictureTiago Hands@tiago_handsOct 14
📐 Geometry Insights 📚 Discover more about GCSE Higher Mathematics, A-Level Mathematics and A-Level Further Mathematics. 🎓 Learn how Geometry Insights aims to bridge the gap between school, college and university mathematics. 🔗 geometryinsights.com/gcse_an…
GCSE and A-Level Examinations

GCSE and A-Level Examinations

Explore GCSE and A-Level Mathematics through rigorous proofs, derivations and diagrams, bridging the gap between school, college and university mathematics.

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📐 Premium Derivatives — Geometry Insights 27 articles covering first principles, core rules, exponential, logarithmic, trigonometric, hyperbolic and inverse-function derivatives. Rigorous workings. Clear conditions. 🔗 geometryinsights.com/derivat…
Derivatives and Differentiation: Rules, Proofs and Derivations

Derivatives and Differentiation: Rules, Proofs and Derivations

Explore derivative and differentiation articles covering first principles, core rules, trigonometric, hyperbolic, exponential, logarithmic and inverse-function derivatives.

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Latest articles on Geometry Insights:
Tiago Hands (@tiago_hands) profile pictureTiago Hands@tiago_handsOct 10
📐 New on Geometry Insights A fresh run of articles on inverse trigonometric and hyperbolic differentiation — from arcsin, arccos and arctan through to coth, cosech, sech and arsinh. Explore the newest articles here: 🔗 geometryinsights.com/article…
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Image from Tiago Hands (@tiago_hands)
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📚 346 mathematics articles — and growing. Explore Geometry Insights & Geometric Bites, with both FREE and PREMIUM articles covering geometry, algebra, calculus, vectors, matrices and more. 🔓 Join: geometricbites.com/p/article… 📐 Free vs Premium: geometricbites.com/p/free-vs…
Image from Mathematics Proofs: GCSE & A-Level (@mathsproofs)
Image from Mathematics Proofs: GCSE & A-Level (@mathsproofs)
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229 articles and counting on Geometry Insights. You can now browse by article number and see what has been covered.
Tiago Hands (@tiago_hands) profile pictureTiago Hands@tiago_handsOct 6
📚 Geometry Insights now has a live Articles by Number index. 🔢 Browse the full archive in publication order ⚡ Updates automatically as new articles are published 📄 30 articles per page 💾 Export the complete list as CSV or JSON 🔗 geometryinsights.com/article…
Articles by Number

Articles by Number

Browse the complete Geometry Insights article library by publication number, with 30 articles per page and automatic updates as new mathematics articles are published.

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Mathematics Proofs: GCSE & A-Level retweeted
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Tiago Hands@tiago_handsOct 6
Pythagoras' Theorem: Detailed Diagram ⏳Time saving tools 06.10.2026 #pythagorastheorem
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Mathematics Proofs: GCSE & A-Level retweeted
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Tiago Hands@tiago_handsOct 5
📐 Locus of the Midpoint of the Axis Intercepts of a Tangent to an Ellipse A and B are the moving axis intercepts of the tangent. P is their midpoint, tracing out the locus as the tangent moves around the ellipse.
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Mathematics Proofs: GCSE & A-Level retweeted
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Tiago Hands@tiago_handsOct 4
📐 Why does every point on an ellipse satisfy PS + PS′ = 2a? This geometric proof uses the foci, directrices, eccentricity and a detail often skipped in proofs: handling absolute values correctly. 🔍 geometryinsights.com/2026/10…
Geometric Proof That PS + PS′ = 2a for the Ellipse x²/a² + y²/b² = 1

Geometric Proof That PS + PS′ = 2a for the Ellipse x²/a² + y²/b² = 1

Prove geometrically that PS + PS′ = 2a for an ellipse using its foci, directrices, eccentricity, vector distances and absolute values.

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Mathematics Proofs: GCSE & A-Level retweeted
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Tiago Hands@tiago_handsSep 28
📐 A simple hyperboloid in ℝ³ from a hyperbola in ℝ²: x² − y² = 1 ⟶ x² + y² = 1 + z² 📘 Full derivation: geometryinsights.com/2026/09… 🌐 3D graph: desmos.com/3d/mrbgwaitql
How to Generate a Simple Hyperboloid x² + y² = 1 + z² in ℝ³

How to Generate a Simple Hyperboloid x² + y² = 1 + z² in ℝ³

Discover how to generate a simple hyperboloid in 3D using mathematical relations and an interactive graph. Learn the basics today!

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Mathematics Proofs: GCSE & A-Level retweeted
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Tiago Hands@tiago_handsSep 28
📐 How can the hyperboloid x² + y² = 1 + z² be parametrised? Start with half a hyperbola in ℝ³, then rotate it through 360° about the z-axis. The result: (sec u cos v, sec u sin v, tan u)ᵀ Full derivation: geometryinsights.com/2026/09…
How to Parametrise the Hyperboloid x² + y² = 1 + z² in ℝ³

How to Parametrise the Hyperboloid x² + y² = 1 + z² in ℝ³

Discover how to parametrize the hyperboloid equation x² + y² = 1 + z² in ℝ³ and visualize it through rotation and graphs.

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Mathematics Proofs: GCSE & A-Level retweeted
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Tiago Hands@tiago_handsSep 28
📐 How do you parametrise a sphere in ℝ³? Start with a semicircle in the plane y = 0, then rotate it through 360° about the z-axis. The full derivation shows the matrix multiplication and verification. geometryinsights.com/2026/09…
How to Parametrise a Sphere in ℝ³

How to Parametrise a Sphere in ℝ³

Discover how to parametrise a sphere in ℝ3 by rotating a semicircle about the z-axis. Explore the math behind this geometric technique!

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Mathematics Proofs: GCSE & A-Level retweeted
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Tiago Hands@tiago_handsSep 29
📘 Article 208 🌀 How is a toroidal helix in ℝ³ actually constructed? The formula is much easier to understand once you know where it comes from. 📐 Article: geometryinsights.com/2026/09… 🧭 Graph: desmos.com/3d/fwhebaglct
How to Construct the Toroidal Helix in ℝ³

How to Construct the Toroidal Helix in ℝ³

Learn to construct a toroidal helix in ℝ³ by using a circle in the y=0 plane and rotating it around the z-axis. Explore the parameters!

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Mathematics Proofs: GCSE & A-Level retweeted
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Tiago Hands@tiago_handsSep 30
📐 New on Geometry Insights: For the hyperbola x²/a² − y²/b² = 1 derive the tangent equation ay sinh t + ab = bx cosh t from the parametrisation x = a cosh t, y = b sinh t. geometryinsights.com/2026/09…
Hyperbola x²/a² − y²/b² = 1: Tangent ay sinh t + ab = bx cosh t

Hyperbola x²/a² − y²/b² = 1: Tangent ay sinh t + ab = bx cosh t

Discover the proof of the tangent to a hyperbola. Learn how to derive the equation using parametrisation techniques.

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Mathematics Proofs: GCSE & A-Level retweeted
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Tiago Hands@tiago_handsSep 30
📐 New on Geometry Insights: For the hyperbola x²/a² − y²/b² = 1 derive the normal at (a sec t, b tan t) to obtain by + ax sin t = (a² + b²) tan t geometryinsights.com/2026/09…
Normal to the Hyperbola x²/a² − y²/b² = 1 at (a sec t, b tan t): by + ax sin t = (a² + b²) tan t

Normal to the Hyperbola x²/a² − y²/b² = 1 at (a sec t, b tan t): by + ax sin t = (a² + b²) tan t

Derive the normal to x²/a² − y²/b² = 1 at (a sec t, b tan t), obtaining by + ax sin t = (a² + b²) tan t.

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Mathematics Proofs: GCSE & A-Level retweeted
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Tiago Hands@tiago_handsOct 1
📐 When is the line y = mx + c tangent to the hyperbola x²/a² − y²/b² = 1? The condition turns out to be: ✨ b² + c² = a²m² See the full derivation from the quadratic discriminant: geometryinsights.com/2026/10…
Condition for y = mx + c to Be Tangent to the Hyperbola x²/a² − y²/b² = 1: b² + c² = a²m²

Condition for y = mx + c to Be Tangent to the Hyperbola x²/a² − y²/b² = 1: b² + c² = a²m²

Derive the condition b² + c² = a²m² for the line y = mx + c to be tangent to x²/a² − y²/b² = 1 using the quadratic discriminant.

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Mathematics Proofs: GCSE & A-Level retweeted
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Tiago Hands@tiago_handsOct 1
📐 New proof on Geometry Insights Why does sin u < u < tan u hold for 0 < u < π/2? Using a unit circle, two triangles and a sector, the inequality falls straight out of geometry. 🔵📏 geometryinsights.com/2026/10…
Geometric Proof of sin u < u < tan u for 0 < u < π/2

Geometric Proof of sin u < u < tan u for 0 < u < π/2

Prove sin u < u < tan u for 0 < u < π/2 by comparing the areas of two triangles and a sector in the unit circle.

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Mathematics Proofs: GCSE & A-Level retweeted
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Tiago Hands@tiago_handsOct 2
📐 Why does a parabola satisfy y² = 4ax? 🧭 Start with its focus and directrix, then use vectors and Euclidean distance to derive the equation directly from the geometry. 🔗 geometryinsights.com/2026/10…
Vector Derivation of the General Equation of a Parabola y² = 4ax in Cartesian Form

Vector Derivation of the General Equation of a Parabola y² = 4ax in Cartesian Form

Derive the parabola equation y² = 4ax from its focus-directrix definition using vectors and Euclidean distance.

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Mathematics Proofs: GCSE & A-Level retweeted
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Tiago Hands@tiago_handsOct 3
📚 Catch up with the latest on Geometry Insights. New geometric proofs, derivations, conic sections, parametrisations, vectors and more — all built to explain why the mathematics works. 📐🧠 Explore the newest articles: geometryinsights.com/recent-…
Recent Posts

Recent Posts

Geometry Insights / Latest Recent Posts The latest mathematical explanations, derivations, proofs and geometric insights published across Geometry Insights. Loading latest articles… Full archive → …

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Mathematics Proofs: GCSE & A-Level retweeted
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Tiago Hands@tiago_handsOct 3
📐 Explore an ellipse through eccentricity. Adjust the parameters and watch the focus, directrix and ellipse change together. 🔍📈 Interactive graph: desmos.com/calculator/fzxkzd… For the full derivation and explanation: geometryinsights.com/2026/10…
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Mathematics Proofs: GCSE & A-Level retweeted
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Tiago Hands@tiago_handsOct 3
📐 How does eccentricity generate an ellipse? Starting from focus S = (ae, 0) and directrix x = a/e, derive the standard equation — then obtain its parametrisation naturally. 🔍📈 geometryinsights.com/2026/10…
Deriving the Standard Equation and Parametrisation of an Ellipse from the Focus–Directrix Definition

Deriving the Standard Equation and Parametrisation of an Ellipse from the Focus–Directrix Definition

Derive the standard equation and parametrisation of an ellipse from the focus–directrix definition of eccentricity, with a rigorous step-by-step proof and interactive graph.

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