SCHOOL OF MATHEMATICS AND STATISTICS MATH3511 Transformations, Groups and Geometry Term 3, 2019 Assignment 1 Due 11pm, 27th October 2019 on moodle The first three questions may be submitted as a group response, or individually. Groups may consist of at most 5 students: each student in the group must submit the same files for the answers to those questions and each file must list the students in the group by full name. Each student in the group will get the same mark for these three questions. Note that this does not excuse you from the University’s standard plagiarism regulations. The last question depends on each student’s student ID number and so must be sub- mitted as a separate file. 1. [7 marks] In △ABC let A′, B′ and C ′ be the midpoints of sides BC, CA and AB respectively. Also, let D on BC, E on CA and F on AB be points such that the cevians AD BE and CF are concurrent at P . Define three new points X = B′D ∩ AB, Y = C ′E ∩BC and Z = A′F ∩AC. Prove that X , Y and Z are collinear. 2. [12 marks] Suppose that for △ABC you are given A and the two Euler points Ea and Eb (you may assume these three points are not collinear). Prove it is possible to find the other two vertices B and C. 3. [6 marks] Suppose three circles are tangent to one another in pairs at three distinct points (which may or may not form a triangle), see figure 1 overleaf. By considering what happens when you invert all three circles in a suitable circle whose centre is one of the points of contact, or otherwise, prove there are exactly two circles that are simultaneously tangent to all three circles, and these two circles do not intersect. (You may assume without proof that inversion preserves tangency.) 4. [5 marks] Suppose your student number is n1n2n3n4n5n6n7. Let X be the point (n1,−n7), Y the point (−n2, n6) and Z the point (n3 − 1, n4 − n5). (If these three points are collinear, replace Z with (n3, n4 − n5).) Find an affine transformation that maps the origin to X , the point (1, 0) to Y and (0, 1) to Z. Is your map direct or indirect? Is it an isometry? (Explain your answers.) 1 Figure 1: Question 3 2