Mathematics Proofs: GCSE & A-Level

Pythagoras' Theorem: Detailed Diagram ⏳Time saving tools 06.10.2026 #pythagorastheorem

📐 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.

📐 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…

📐 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 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 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…

📘 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

📐 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…

📐 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…

📐 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…

📐 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…

📐 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…

📚 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-…

📐 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…

📐 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…

📐 Geometry Insights has reached Article 202 — and the collection keeps growing. Recent articles focus on how objects in ℝ³ are actually constructed, not just the final result. Learn the constructions here: geometryinsights.com/recent-…

🌀 Article 202: How to Generate a Helicoid in ℝ³ Start with two opposite points on rotating circles in ℝ³, join them parametrically, and simplify the resulting vector equation to generate the helicoid. 🔗 geometryinsights.com/2026/09… #Geometry #Maths #Helicoid

🌀 How do you turn a circle in ℝ² into a helix in ℝ³? Add vertical motion to circular motion — then control the radius, rotation rate and visible length. 🔗 geometryinsights.com/2026/09… #Geometry #Maths

📐 Article 200 on Geometry Insights! To celebrate: how do you generate a Pringle mathematically? Start with a hyperbolic paraboloid in ℝ³ and restrict it to a finite elliptical region. 200 articles of exploring the geometry behind the equations. 🔗 geometryinsights.com/2026/09…

📐 From x² − y² = (x + y)(x − y) to a hyperbolic paraboloid in ℝ³. 📖 Article: geometryinsights.com/2026/09… 📊 Parametrised: desmos.com/3d/idxetl6any 🌐 Surface: desmos.com/3d/gddmgiv0zd