Pleasing Pictorial Proofs and Ptolemys Ptheorem Martin Harris

Pleasing Pictorial Proofs and Ptolemy's Ptheorem Martin Harris @Mar. Har. Star Maths. Jam Annual Conference 2018

Pictorial Proof of Pythagoras a² c² b²

Pictorial Proof of Cosine Rule b a a b s. C co. b a ab. co s. C A s o . c c b C ac. co a B c ac. cos. B c²=a²+b²-2 ab. cos. C s. B A bc. cos. A c c b

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

Refresher on Circle Theorems α 1) α 3) sin 2) 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° α α 180°-α 1

Pythagoras’ Theorem b a c c b a

Double Angle Formulae (I) sin. A. sin. B A A B B A cos. A. cos. B sin. A. cos. B B s co 1 cos B A sin(A+B) sin A s o c sin B B cos. A. sin. B sin 1 B) + A ( n i s cos(A+B) A+B

Double Angle Formulae (II) sin. A. sin. B B A A cos. A. cos. B A 1 B 90°-(A+B) A A B sin. A. cos. B B s co ) B + A ( s co A s o c cos B sin(A+B) cos. A. sin. B sin 1 sin B cos(A+B) A+B B

My General Proof α+β=θ 1 (both are 180° - θ 2) If we flip the bottom section, the new quadrilateral has previously opposite sides now adjacent, with θ angles between them α Q P β θ 2 S We can then cut along the dotted line, and make two copies of each triangle into the required parallelograms θ 1 α R P Q θ 2 θ 1 R S

Final Proof P Q P X θ 2 θ 1 S R x 2 Q θ 1 S θ 2 θ 1 Y x 2 X θ 1 Y θ 2 R θ 2

Thank You Desmos: https: //www. desmos. com/calculator/6 ebufxblaa @Mar. Har. Star

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

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