E-320: Teaching Math with a Historical Perspective Oliver Knill, 2012Lecture 3: WorksheetsThales theoremIt is a result of Thales of Miletus (625 BC -546 BC) stating that if a triangle inscribed in afixed circle is deformed by moving one of its points on the circle, then the angle at this pointdoes not change. The result is relevant also because the discoverer Thales is considered the firstmodern Mathematician. Thales theorem is a prototype of a stability result. We look at a slightlymore general case than usual treated which is called the ”Fass kreis” theorem in Europe. In thisworksheet we want to understand it and prove it.Lets look first at the case when one side of the triangle goes through the center.a) The triangle BCO is an isosceles triangle.b) The central angle AOB is twice the angle ACB.OCBA11ODCBAc) What is the relation between the angles AOD and ACD?d) What is the relation b etween the angles DOB and DCB?e) Find a relation between the central angle AOB and the ang le ACB?f) Why does the angle ACB not change if C moves on the circle?In this worksheet, we prove the quadrature of the Lune, a result of Hippocrates of Chios (470BC - 400 BC). It is the first rigorous quadrature of a curvilinear area.2. Hyppocrates theoremThe sum L + R of the area L of the left moon and the area R of the right moon isequal to the area T of the triangle.RLTa) Relate first the a r ea of each half circle with the corresponding triangle side length. Let nowA, B, C the areas of the half circles build over the sides of the triangle. Show that A + B = C.The picture shows the half disc below the hypothenuse.BACb) Let U be the area of the intersection of A with the upper half circle C. and let V be the areaof the intersection of B with C. Let T be the area of the triangle. Why is U + V + T = C?c) The number A −U is the area of a region. Which one. Similarly, what is B − V ?Can you finish the proof of the theorem L + R = T ?VUTCThe butterfly theoremWe want to see why the butterfly wings in the butterfly theorem are similar triangles:Morley’s miracleWe want to understand the pr oof o f Morley’s miracle and build the triangle up with 7 triangles.ΑΑΑΒΒΒΓΓΓFasskreis theoremGiven a circle of radius 1 and a p oint P inside the circle. For any line through P which intersectsthe circle at points A, B we have |P O|2− |P A||P B| = 1.Pro of with Pythagoras. By scaling translation and rotation we can assume the circle is at theorigin and t hat the line through the point P = (a, b) is horizontal. The intersection points a rethen (±√1 −a2, a). Now (b −√1 − a2)(b +√1 − a2) = b2− 1 +