SCHOOL OF MATHEMATICS AND STATISTICS MATH3511 Transformations, Groups and Geometry Term 3, 2019 Assignment 2 Due 11pm, 15th Nov 2019 on moodle 1. [7 marks] In 4ABC let Ra be reflection in the internal bisector at A and similarly define Rb and Rc. Prove that the composition T = Rc ◦ Rb ◦ Ra is a reflection in the line through the incentre I and the point X where the incircle touches AC. 2. [6 marks] Let R be a reflection in line ` and T a glide reflection that leaves a line λ orthogonal to ` invariant. Prove that the compositions T ◦ R and R ◦ T are half turns. Where is the fixed point of the half turns (with respect to the intesection of the lines ` and λ)? 3. [8 marks] Suppose the three triangles T1 = 4A1B1C1, T2 = 4A2B2C2 and T3 = 4A3B3C3 are all in perspective from a point O (so that O, A1, A2, A3 are collinear etc). By Desargues theorem there are lines L12, L13 and L23 such that T1 and T2 are in perspective from L12, T1 and T3 are in perspective from L13 and T2 and T3 are in perspective from L23. Prove that L12, L13 and L23 are concurrent. Note: your major issue here may be finding a sensible notation! 4. [5 marks] Let T : RP2 → RP2 be a projective transformation. Show that if T fixes all the points at infinity it is either a translation or a radial transformation. 5. [4 marks] Suppose your student number is n1n2n3n4n5n6n7. Find the equations of the lines through points with homogeneous coordinates (a) [n2, n4, n6] and [−1, 1, 1] and (b) [n3, n5, 1] and [−2, 1, 0] in RP2 and find the homo- geneous coordinates of their intersection. Give the homogeneous coordinates of the ideal point on the first of these lines. (You may use the result of Q84 on the problem sheets without needing to prove it.) 1
