Euclidean and non-euclidean Geometry 2018

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1 r. ernd Kreussler Mini rojects MH uclidean and non-uclidean Geometry xternal tangents to two unequal circles (week 8) Given two circles of different radius not containing each other. xplain the construction of the common external tangents sketched below and justify the construction. G I O M J H Hint: =, M is the mid-point of O, M = MO = M and GI =. 2. ick s Theorem (week 8) The points in the plane with integer co-ordinates are known as the lattice points. lattice polygon is a polygon with all vertices having integer co-ordinates. We only consider lattice polygons the boundary of which does not have self-intersections. Let i be the number of lattice points inside a given lattice polygon and b the number of lattice points on its boundary. rove that the area of such a lattice polygon is equal to i+ b 2 2. References: [OW] p.278, 279 (Varberg s proof, 1985) and p tolemy s Theorem (week 9) Leta,b,c,darethesidelengthsandd 1,d 2 thelengthsofthediagonalsofacyclicquadrilateral. rove that ac+bd = d 1 d 2. References: [OW] p.114, Miquel s Triangle Theorem (week 9) Let, and be points on the sides,,, respectively, of triangle. rove that the circumcircles of the triangles, and meet at a point M. References: [OW] p.95. b M c d d 1 d 2 a Miquel s Triangle Theorem tolemy s Theorem

2 r. ernd Kreussler Mini rojects MH The Nine oint ircle (week 9) rove that, in any triangle, the following nine points lie on one circle: the feet of the three altitudes; the three mid-points between the orthocentre and the vertices; the three mid-points of the sides. References: [] p.18, The uler Line (week 9) Let beatriangleandh,gando betheorthocentre, thecentroidandthecircumcentre, respectively. rove that H,G,O are on a line (the uler Line) and that HG = 2 GO. References: [] p The Simson Line (week 10) Let be a point on the circumcircle of triangle, and let, and be the feet of the perpendiculars from to the (possibly produced) sides of the triangle. rove that, and are on a line. References: [OW] p.217, The Torricelli-ermat oint (week 10) Let, and be equilateral triangles constructed outwards at the sides of a triangle in which all angles are less than 120. rove that the lines, and meet at a point which is on the circumcircle of each of the triangles, and. References: [OW] p.109, 110, x 9(a) and (b), Sol p.359, uler rahmagupta ormula (week 10) rove (without using Heron s ormula) that the area of a cyclic quadrilateral with sides a,b,c,d is equal to (s a)(s b)(s c)(s d) where s = (a+b+c+d)/2 is the semi-perimeter. References: [OW] p.182 x 13, Sol p.373, aided by p.172, 174, 175.

3 r. ernd Kreussler Mini rojects MH Orthic Triangle (week 10) Let, and be the feet of the altitudes of triangle. The triangle is called the orthic triangle. rove that the triangles, and are all similar to. Use this to show that = acosα, = bcosβ and = ccosγ. References: [OW] p and p.153 x 14, Sol p Golden Section (week 10) xplain the Golden Section. xplain the construction sketched below and prove that the point G is the Golden Section of the initially given line segment. Hint: The centres of the circles and arcs are,, and. ind the length of from and then use the osine Rule for G. G Golden Section onstruction of 1/8 12. onstruction of 1/8 (week 10) xplaintheconstructionaboveinwhich and weregiveninitially. rove that = 1. 8 Hint: The line is the radical axis of two intersecting circles. 13. ircle of pollonius (week 10) Two points, and a number λ > 1 are given. rove that the set of all points that satisfy = λ is a circle. References: [] p.88, Radical xis (week 10) xplain what the radical axis (line of equal powers) of two non-intersecting circles is and why it is a line. What is the radical centre of three circles? rove that the radical centre of any three circles exists. xplain (with proof) how to construct the radical axis of two circles that do not intersect. References: [OW] p.99, 100 and en.wikipedia.org/wiki/radical axis.

4 r. ernd Kreussler Mini rojects MH Van Schooten s Theorem (week 11) Let be a point on the circumcircle of an equilateral triangle such that intersects the line segment. rove that = +. Hint: onsider points, on for which = and =, and then compare the triangles and. Q X M Y Van Schooten s Theorem utterfly Theorem 16. utterfly Theorem (week 11) Through the midpoint M of a chord Q of a circle, two other chords and are drawn such that chords and meet Q at points X and Y. rove that M is the midpoint of XY. References: [G] Theorem 2.81, p Generalised arallelogram Law (week 11) rove the following formula of uler (1750) which is valid for any convex quadrilateral with sides a,b,c,d and diagonals of lengths d 1 and d 2. The number m in the formula denotes the distance between the midpoints of the two diagonals. a 2 +b 2 +c 2 +d 2 = d 2 1 +d2 2 +4m2 b c d 2 d 1 m a d References: [OW] p.233 x 3, Sol p.377(you should use vectors to simplify their calculations). 18. ngle isector Length (week 11) Let w be the length of the angle bisector through in triangle. rove that References: [OW] p w cos γ 2 = 1 a + 1 b.

5 r. ernd Kreussler Mini rojects MH argeter s ormula (week 11) Let be a convex quadrilateral and let be the intersection point of the diagonals and. Using the notation α =, β =, γ = and δ = as indicated in the diagram, prove that = cotα+cotβ cotγ +cotδ. δ γ β α References: [a] p.218 and the formula (m+n)cotθ = mcot ncot, see [u] p Values of trigonometric functions (week 11) Without using a calculator, prove that 5 1 sin18 =, cos18 = References: [OW] p.150, 151 x 4, Sol p.362, o-ordinates of ircumcentre (week 11) and tan18 = Let triangle be placed in such a way in a artesian co-ordinate system that is at the origin and is at distance c from on the positive x-axis. rove that the artesian co-ordinates of the circumcentre O are ( ) c 2, c(a2 +b 2 c 2 ). 8area() References: [OW] p.214, 215, 120 (ig. 5.9 (left)). 22. uler s ormula (week 11) rove that π = arctan 1 +arctan 1. Use this formula and the first three or four terms of the Taylor series for arctan(x) to calculate an approximation for π. References: [OW] p.289, Spherical Triangles (week 12) or a spherical triangle with side lengths a,b,c, we let s = (a+b+c)/2. rove that ( α sin(s b)sin(s c) sin =. 2) sin(b) sin(c) References: [OW] p.154, x 22, Sol. p.367 and p.118, 133.

6 r. ernd Kreussler Mini rojects MH Stereographic rojection (week 12) xplain the definition of the stereographic projection from the sphere to the plane and prove that it preserves angles. References: [OW]: p Hyperbolas (week 12) Starting with the geometric definition of a hyperbola (given two foci and a constant a), prove that one branch of it can be described in polar coordinates by the equation r = ed 1 ecos(ϕ). xplain the choice of the coordinates, the possible values of e and ϕ, and what the directrix of a hyperbola is. References: Your lecture notes week The izza Theorem (week 12) circular pizza is cut into eight pieces by four concurrent straight lines at equal angles to each other. The common intersection point of these four lines is inside the pizza but does not need to be its centre. rove that the sum of the areas of the odd-numbered pieces is equal to the sum of the areas of the even-numbered pieces. Hint: oryourproofyouneedtousethattheregionthatiscutoutbethecurver = r(ϕ)and b a r(θ)2 dθ. You two line segments connecting its end points to the origin has area equal to 1 2 need to prove this formula using Riemann Sums or other tools from your calculus modules. You also need to prove the following our-squares-theorem: If two chords of a circle intersect at right angles forming four segments a,b,c,d, then a 2 +b 2 +c 2 +d 2 = 4r 2, where r is the radius of the circle. To prove the izza Theorem you will use polar coordinates centred at the point where the four lines intersect inside the izza r c a b d References: [OW] p.26 x 24, Sol p.349 (our-squares-theorem). References [] H.S.M. oxeter: Introduction to Geometry, John Wiley, 2 nd d, [G] H.S.M. oxeter, S.L. Greitzer: Geometry Revisited, M, [u] H.M. undy: eedback on 88.15, The Mathematical Gazette, Vol 88, No 513, pp , [OW]. Ostermann, G. Wanner: Geometry by its History, Springer, [a].r. argeter: Reflections upon a Theme, The Mathematical Gazette, Vol 48, No 364, pp , 1964.

Mathematics 10 Page 1 of 6 Geometric Activities

Mathematics 10 Page 1 of 6 Geometric ctivities ompass can be used to construct lengths, angles and many geometric figures. (eg. Line, cirvle, angle, triangle et s you are going through the activities,