MATH3511代做、代写STATISTICS、代做Python编程设计、c/c++，Java代写代写

日期：2019-10-29 09:43

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 submitted

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.)

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Figure 1: Question 3

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