Mathematics Proofs: GCSE & A-Level

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[Mathematics Proofs: GCSE & A-Level @mathsproofs]

More information about GCSE and A-Level Examinations in the UK and how Geometry Insights aims to help students.

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📐 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.

🔗 https://t.co/oSXyf6J4Ra

[Mathematics Proofs: GCSE & A-Level @mathsproofs]

📐 Premium Derivatives — Geometry Insights

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

Rigorous workings. Clear conditions.

🔗 https://t.co/ySOp1GRG4Y

[Mathematics Proofs: GCSE & A-Level @mathsproofs]

Latest articles on Geometry Insights:

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📐 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:

🔗 https://t.co/HmeYJflaWG

[Mathematics Proofs: GCSE & A-Level @mathsproofs]

📚 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: https://t.co/da99QmuCPj

📐 Free vs Premium: https://t.co/skldVMEfPw

[Mathematics Proofs: GCSE & A-Level @mathsproofs]

229 articles and counting on Geometry Insights. You can now browse by article number and see what has been covered.

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📚 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
🔗 https://t.co/HmeYJfkD78

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Pythagoras' Theorem: Detailed Diagram

⏳Time saving tools

06.10.2026

#pythagorastheorem

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📐 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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📐 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. 🔍

https://t.co/P1EZOQZzx5

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📐 A simple hyperboloid in ℝ³ from a hyperbola in ℝ²:

x² − y² = 1 ⟶ x² + y² = 1 + z²

📘 Full derivation:
https://t.co/QY41bl3pfo

🌐 3D graph:
https://t.co/HJJDnjR0sp

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📐 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:
https://t.co/DDqf9pJ3Oy

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📐 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.

https://t.co/FR8B2SM3gU

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📘 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:
https://t.co/zRjASdMrOK

🧭 Graph:
https://t.co/C9nSRy73E2

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📐 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.

https://t.co/Bh00zoLEWf

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📐 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

https://t.co/bxmuDfBCSZ

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📐 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:

https://t.co/gFv8qeOY4Y

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📐 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. 🔵📏

https://t.co/L04ex7nGUw

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📐 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.

🔗 https://t.co/9AtHnte2ZD

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📚 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:
https://t.co/uVA2kZP87a

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📐 Explore an ellipse through eccentricity.

Adjust the parameters and watch the focus, directrix and ellipse change together. 🔍📈

Interactive graph:
https://t.co/OB7YwpY6fp

For the full derivation and explanation:
https://t.co/skvYuJotmw

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📐 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. 🔍📈

https://t.co/skvYuJotmw