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Pleasing Pictorial Proofs and Ptolemy's Ptheorem

Explore visual proofs of Pythagoras’ Theorem and Ptolemy’s Theorem, with additional insights into Circle Theorems and the Golden Ratio. This presentation provides a refreshing take on geometric concepts through pictorial representations and step-by-step explanations.

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Pleasing Pictorial Proofs and Ptolemy's Ptheorem

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  1. Pleasing Pictorial Proofs and Ptolemy's Ptheorem Martin Harris @MarHarStar MathsJam Annual Conference 2018

  2. Pictorial Proof of Pythagoras a² c² b²

  3. Pictorial Proof of Cosine Rule b b ab.cosC a bc.cosA a C ab.cosC b ac.cosB a B A ac.cosB bc.cosA c²=a²+b²-2ab.cosC c c c

  4. Ptolemy’s Theorem: The product of the diagonals of a cyclic quadrilateral is equal to the sum of the products of opposite pairs of sides. P X Y Q S R PR + QS = XY

  5. Refresher on Circle Theorems α α • Angles subtended by the same chord are the same angle • On a unit diameter circle, the chord subtended by angle α has length sin α • Opposite angles add up to 180° sinα α 180°-α 1

  6. Pythagoras’ Theorem b a a c c b

  7. Double Angle Formulae (I) cos(A+B) sinA.sinB A+B sin B cosA.sinB 1 cos A sin A sin(A+B) sin(A+B) A 1 B sin B cos B cos B sinA.cosB B A A cosA.cosB

  8. Double Angle Formulae (II) cos(A+B) sinA.sinB A+B sin B cosA.sinB cos(A+B) 1 cos A sin A cos B sin B sin(A+B) B A 1 cos B sinA.cosB B A B A A 90°-(A+B) cosA.cosB

  9. Golden Ratio: φ²=φ+1 1x1 1×(1/φ) φ φ φ² φ×1 φ φ 1 1 1 1×(1/φ) 1

  10. My General Proof α α+β=θ1 (both are 180° - θ2) Q P Q P If we flip the bottom section, the new quadrilateral has previously opposite sides now adjacent, with θ angles between them θ1 β θ2 θ2 θ1 α S S R R We can then cut along the dotted line, and make two copies of each triangle into the required parallelograms

  11. Final Proof P x2 Q P Q X θ2 S R θ2 θ2 Y θ1 θ1 θ1 θ1 S R x2 X θ2 Y

  12. Thank You Desmos: https://www.desmos.com/calculator/6ebufxblaa @MarHarStar

  13. A Nice Special Case QS cos²β α β PR cos²α P cosα S cosβ P S Q cosβ R cosα β α P sinα S sinβ R sinα Q sinβ QS sin²β Q R PR sin²α

  14. An Unsatisfying (but Technically Correct) Proof QS PR Q(PSX) P(QRX) Q P X PQ XQ PQ XP Y X(PQY) S R XY Line lengths are not scalar factors!

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