Heron’s Formula for Triangular Area

1. Heron’s Formula for Triangular Area April Gordon Denise Hunter Ha Nguyen Math 3031 5 February 2009

2. Math History Eratosthenes ca. 230 BC World History

3. Eratosthenes (ca.284-192 BC) • Greek mathematician, astronomer, geographer • Chief librarian of the Library of Alexandria

4. Eratosthenes (ca.284-192 BC) • Circumference of the Earth Syene to Alexandria 7.2 ˚ -------------------------------- = -------- Earth’s circumference 360 ˚ Eratosthenes’ estimate: 24,466 miles Accepted value: 24,860 miles

5. Eratosthenes (ca.284-192 BC) • Also known for • Mapping of the world according to longitude and latitude • Divided the earth into climatic zones • Prime sieve • Poem “Hermes”

6. Math History Apollonius Archimedes Eratosthenes ca. 230 BC ca. 225 BC ca. 210 BC World History Han dynasty Great Wall of China Qin dynasty ca. 221 BC ca. 202 BC

7. Apolloniusof Perga (ca.262-190 BC) • “The Great Geometer” • Conics

8. Math History Apollonius Hipparchus Archimedes Eratosthenes ca. 230 BC ca. 225 BC ca. 210 BC ca. 150 BC World History Han dynasty Great Wall of China Qin dynasty ca. 221 BC ca. 202 BC

9. Hipparchus (ca. 190 -120 BC) • First person documented to use trigonometry • Chord table • Catalogue of over 850 fixed stars

10. Math History Liu Hsin Heron Apollonius Hipparchus Archimedes Posidonius Eratosthenes ca. 230 BC ca. 225 BC ca. 210 BC ca. 150 BC ca. 100 BC ca. 1 AD ca. 75 AD World History Caesar assassinated Romans destroy Carthage Trade along Silk Road Han dynasty Colosseum Roman’s take Egypt Roman Aqueducts Great Wall of China Qin dynasty ca. 221 BC ca. 202 BC ca. 146 BC ca. 110 BC ca. 44 BC ca. 30 BC ca. 79 AD

11. Heron of Alexandria • Also known as Hero • Mathematician, physicist and engineer • Taught at Museum of Alexandria (ca. 75 AD ?)

12. Some works of Heron • Mechanics • Mechanical machines, methods of lifting • Dioptra • Surveying, instruments for surveying • Pneumatica • Describes various types of machines and devices • Metrica • Most important geometric work, included methods of measurement

13. Pneumatica • Earliest known slot machine • Sacrificial Vessel which flows only when money is introduced. • Automatic opening of temple doors • Temple Doors opened by fire on an altar.

14. Aeolipile • “Wind ball” in Greek • Earliest recorded steam turbine • Regarded as a toy • Principle similar to jets

15. Metrica • Areas of triangles, polygons, surfaces of pyramids, spheres, cylinders • Volumes of spheres, prisms, pyramids • Divisions of areas and volumes in parts

16. Approximating a square root • Heron’s method for the square root of a non square integer • If , is approximated by • Successive approximation gives better results • ie. If is the first approximation for is a better approximation, but is even better and so on.

17. Great Theorem: Heron’s Formula for Triangular Area • Why? • Uses SSS congruence • No intuitive appeal • Formula: a b ? c where

18. Propositions Leading to Heron’s Formula • The bisectors of the angles of a triangle meet at a point that is the center of the triangle’s inscribed circle.

19. Propositions • The bisectors of the angles of a triangle meet at a point that is the center of the triangle’s inscribed circle. • In a right-angled triangle, if a perpendicular is drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle and to one another.

20. Propositions • The bisectors of the angles of a triangle meet at a point that is the center of the triangle’s inscribed circle. • In a right-angled triangle, if a perpendicular is drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle and to one another. • In a right triangle, the midpoint of the hypotenuse is equidistant from the three vertices. B M D A C

21. Propositions • The bisectors of the angles of a triangle meet at a point that is the center of the triangle’s inscribed circle. • In a right-angled triangle, if a perpendicular is drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle and to one another. • In a right triangle, the midpoint of the hypotenuse is equidistant from the three vertices. • If AHBO is a quadrilateral with diagonals AB and OH and if <HAB and <HOB are right angles, then a circle can be drawn passing through the verticiesA, O, B, and H. O A B H

22. Propositions • The bisectors of the angles of a triangle meet at a point that is the center of the triangle’s inscribed circle. • In a right-angled triangle, if a perpendicular is drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle and to one another. • In a right triangle, the midpoint of the hypotenuse is equidistant from the three vertices. • If AHBO is a quadrilateral with diagonals AB and OH and if <HAB and <HOB are right angles, then a circle can be drawn passing through the vertices A, O, B, and H. • The opposite angles of a cyclic quadrilateral sum to two right angles.

23. The Theorem • For a triangle having sides of length a, b, and c and area K, we have where is the triangle’s semi-perimeter. • PROOF:ABC is an arbitrary triangle configured so that side AB is at least as long as the other two

24. PROOF: Part A s is the semiperimeter

25. Purpose of part b: to construct the quantities of our interest i.e. r, s, (s – a), (s – b), (s – c) inside the triangle.

26. ∆ OCE = ∆ OCE (sas) • ∆ OBE = ∆ OBD (sas) • ∆ OAF = ∆ OBD (sas) • Then extend BA such that BG = s • From these triangle congruence we have • s – c = CE = CF = AG • s – b = BD = BE • s – a = AD = AF

27. From part a, we have that the area of ∆ ABC is r.s. Need: • rs = √s(s - a)(s - b )(s - c) • r²s² = s(s – a)(s – b)(s – c) • r²s = (s – a)(s – b)(s – c) • r²/ (s – b) = (s – a)(s – c)/ s (1) • From ∆ KOB, we have that OD² = DK.DB, so r² = DK(s – b) • so r²/ (s – b) = DK. (2)

28. Equivalently, need: (s – a)(s – c) = DK.s if so AD.AG = DK.BG (3) • if so AD/ DK = BG/ AG (4) • if so AD/DK – 1 = BG/AG – 1 • then AK/ DK = AB/AG (5) • Now it boils down to prove that • (i) ∆ HAK ~ ∆ ODK • (ii) ∆ OCE ~ ∆ AHB • (iii) angle ABH = ½ (angle ACB)

29. Indeed: • (iii) angle ABH = ½ (angle ACB) by proposition of two opposite angles in a cyclic quadrilateral • (ii) ∆ OCE ~ ∆ AHB (a.a.a) • (i) ∆ HAK ~ ∆ ODK (a.a.a) • So: CE/AB = OE/ AH • CE = AG, OE = OD • Hence AG/ AB = OD/ AH.

30. Now if we shuffle the steps we just went through … we realized that Heron’s proof utilizes many things about geometry, especially cyclic quadrilateral, triangle and circles, triangle congruence and similarity. • But there are more straightforward derivations.

31. Consider the general triangle. • By Pythagorean theorem, b² = h² + u², c² = h² + v² so u² - v² = b² - c² • Dividing both sides by a = u + v … • Adding u + v = a to both sides and solving for u gives u = (a^2 + b^2 - c^2 u)/ 2a • Now just take h = √(b² - u²) …

32. What happens if we factor things inside the square root? Brahmagupta (620 AD) generalized the case beautifully by adding a 4th side: • What happens if we factor out the term ab?

33. This equation is the building block for the third proof: • Which is …?

34. Centers of Mathematical Discovery Rome China Greece Ancient Babylonia Egypt India Arabia

35. B C The World After Heron A Nine Chapters on the Mathematical Art (Jiu Zhang Suanshu) - arithmetic and elementary algebra Al-Battani - bsin(A) = asin(90o-A) Ja’far Muhammad ibn Musa al-Khwarizmi - algebra and algorithms D Greek Trigonometrist and geometer - first to recognize that curves were analogues of straight lines ZhoubiSuanjing – created a visual proof for the Pythagorean Theorem Xiahou Yan used zero as a placeholder Brahmagupta – One of the first to use negative numbers, described how to sum a series, created the rules for zero Aryabhata I - solved basic algebra equations Ex. by = ac + c and by = ax – c where a,b,c are all integers Wang Xiaotong – solved the cubic equation Pappus – developed theorem on volume of a solid of revolution SunziSuanjing - 220 – 473- important book of problems: Ex. A woman aged 29 is 9 months pregnant. What sex is her baby? Mathematics Hypatia - the first notable woman mathematician Abu KamilShuja – link between Arab and European math Ex. x5 = x2x2x and x6=x3x3 100 200 300 370 395 473 475 505 598 625 700 780 800 930 Theon of Smyrna- number theory and mathematic in music 70 Diophantus – father of algebra Claudius Ptolemaeus- famous theorem: Menelaus Pappus Hypatia SunziSuanjing Aryabhata I Al-Battani/ Abu KamilShuja Nine Chapters/ Theon of Smyrna/ Ptolemy ZhoubiSuanjing/ Diophantus Varahamihira Brahmagupta Wang Xiaotong Zero invented Ja’far Muhammad Eruption of Mt. Vesuvius destroyed cities of Pompeii and Herculaneum 15 Jul 622 - Muslim calendar is invented Political division into the Western and Eastern Roman Empires as Christianity becomes the official religion of Rome Tang Dynasty – period of high scholarship Gives freedom of Religion in the Roman empire as the Emperor Constantine I converts to Christianity World History 180 220 312 376 395 438 518 565 570 600 622 641 930 First of the Gothic Wars signaling the collapse of the Roman Empire 79 Gothic Wars begin Mt. Vesuvius eruption Later Roman Empire Sui Dynasty Later Roman Empire ends Tang Dynasty Edict of Religious Toleration Mohammed Han Dynasty ends Christianity-Rome Muslim calendar Alexandrian Library burning Theodosian Code