Course Problems
Home
Problems
Assign Problems
Organize
Assign Problems
Add Problems
Solution Progress
TikZ Images
Compare
Difficulty
Banger Rating
PDF Management
Ctrl+S
Edit Problem
Year
Paper
Question Number
Course
-- Select Course --
6th Form
LFM Pure
LFM Pure and Mechanics
LFM Stats And Pure
UFM Additional Further Pure
UFM Mechanics
UFM Pure
UFM Statistics
zNo longer examinable
Section
-- Select Section --
Geometry
3d Geometry
Algebra
Algebra - Inequalities
Algebra - Quadratic Expressions
Algebra - Simultaneous Equations
Algebra - Number Theory
Problem Solving
Coordinate Geometry
Simultaneous equations
Proof
Inequalities
Proof by induction
Introduction to trig
Modulus function
Matrices
Linear transformations
Invariant lines and eigenvalues and vectors
Trigonometry 2
Small angle approximation
Differentiation
Differentiation - Properties of Differentiation
Differentiation - Analysis of Functions and Inequalities
Differentiation - General Optimization
Differentiation - Geometric Optimization
Integration
Integration - Symmetry, Periodicity, and Special Arguments
Integration - Special Functions and Orthogonal Polynomials
Integration - Advanced & Conceptual Topics
Integration - Cauchy-Schwarz
Implicit equations and differentiation
Differential equations
Differential Equations - Separation of Variables
Differential Equations - Modelling and Applications
Differential Equations - Advanced Systems and Qualitative Analysis
3x3 Matrices
3x3 Matrices - Solving Linear Systems of Equations
3x3 Matrices - Evaluating and Factoring Determinants
3x3 Matrices - Matrices, Vectors, and Geometric Interpretations
Exponentials and Logarithms
Arithmetic and Geometric sequences
Differentiation from first principles
Integration as Area
Vectors
Vectors - Relative Velocity
Constant Acceleration
Non-constant acceleration
Newton's laws and connected particles
Pulley systems
Motion on a slope
Friction
Momentum and Collisions
Moments
Parametric equations
Parametric Equations - Differentiation
Parametric Equations - Integration
Projectiles
Projectiles - Projectile Motion with Geometric Constraints
Projectiles - Conservation of Momentum & Energy in Explosions
Projectiles - Envelope of Trajectories
Quadratics & Inequalities
Curve Sketching
Polynomials
Polynomials - Factor theorem, Remainder Theorem
Polynomials - Repeated Roots and Calculus
Binomial Theorem (positive integer n)
Functions (Transformations and Inverses)
Partial Fractions
Generalised Binomial Theorem
Complex Numbers (L8th)
Complex Numbers - Algebra of Complex Numbers and Polynomials
Complex Numbers - The Geometry of Triangles and Polygons
Complex Numbers - Loci and Regions in the Argand Diagram
Complex Numbers - Complex Mappings and Transformations
Combinatorics
Measures of Location and Spread
Probability Definitions
Tree Diagrams
Principle of Inclusion/Exclusion
Independent Events
Conditional Probability
Discrete Probability Distributions
Uniform Distribution
Binomial Distribution
Geometric Distribution
Hypergeometric Distribution
Negative Binomial Distribution
Modelling and Hypothesis Testing
Hypothesis test of binomial distributions
Data representation
Continuous Probability Distributions and Random Variables
Continuous Uniform Random Variables
Geometric Probability
Normal Distribution
Approximating Binomial to Normal Distribution
Solving equations numerically
Newton-Raphson method
Sequences and Series
Sequences and Series - Solving Recurrence Relations
Sequences and Series - Properties and Identities of Sequences
Sequences and Series - Analysis of Sequences
Number Theory
Vector Product and Surfaces
Groups
Groups - Axioms and Definitions
Groups - Subgroups, Element Order, and Cyclic Groups
Groups - Symmetry, Permutations, and Group Actions
Groups - Matrix Groups
Groups - Isomorphism, Conjugacy, and Structural Properties
Reduction Formulae
Reduction Formulae - Wallis Integrals and Applications
Reduction Formulae - Beta and Gamma-like Integrals
Reduction Formulae - General Trigonometric Integrals
Reduction Formulae - Integrals of Algebraic Functions
Reduction Formulae - Abstract and Iterated Integrals
Moments
Moments - Equilibrium of Rigid Bodies
Moments - Equilibrium of Systems with Multiple Components & Mechanisms
Moments - Friction on Flexible Bodies
Work, energy and Power 1
Momentum and Collisions 1
Momentum and Collisions 1 - Objects entering objects
Centre of Mass 1
Circular Motion 1
Momentum and Collisions 2
Work, energy and Power 2
Centre of Mass 2
Circular Motion 2
Circular Motion 2 - Particle Dynamics in Vertical Circles
Circular Motion 2 - Orbital Mechanics and Central Forces
Circular Motion 2 - Motion in Rotating Frames
Circular Motion 2 - Multi-body Systems and Collisions
Circular Motion 2 - Rigid Body and Rolling Dynamics
Dimensional Analysis
Variable Force
Variable Force - Rectilinear Motion with Resistance
Variable Force - Projectile and 2D Motion with Resistance
Simple Harmonic Motion
SHM - Coupled Oscillators & Normal Modes
SHM - Pendulums: Small Oscillations & Rigid Bodies
SHM - Pendulums: Large Amplitudes & Approximations
SHM - Damped & Forced Oscillations
SHM - Potential Energy & Stability
SHM - Constrained or Non-Inertial Systems
Sequences and series, recurrence and convergence
Sequences and Series - Recurrence Relations and Iterative Sequences
Sequence and Series - The Integral Test and Sum-Integral Bounds
Sequences and Series - Limits and Riemann Sums
Sequences and Series - Summation Techniques and Specific Series
Sequences and Series - Power Series, Taylor Expansions and Differentiation
Sequences and Series - Foundational Concepts: Convergence and Continuity
Sequences and Series - Advanced Proofs and Special Topics
Roots of polynomials
Roots of polynomials - Roots in arithmetic or geometric progression
Roots of polynomials - Transforming Polynomials
Roots of polynomials - Symmetric Functions and Newton's Sums
Roots of polynomials - Special Properties and Conditions on Roots
Polar coordinates
Conic sections
Taylor series
Taylor Series - Series Generation via DEs and nth Derivatives
Taylor Series - Inequalities and Bounds from Taylor Series
Taylor Series - The Remainder Term and Convergence Theory
Taylor Series - Generating Functions in Combinatorics and Algebra
Taylor Series - Limits and Miscellaneous Applications
Hyperbolic functions
Integration using inverse trig and hyperbolic functions
Mean Values
Binary Operations
Vectors
First order differential equations (integrating factor)
Complex numbers 2
Complex Numbers - Geometric Applications of Roots of Unity
Complex Numbers - De Moivre's Theorem and Trigonometric Identities
Complex Numbers - Algebra and Equation Solving
Complex Numbers - Complex Functions, Series, and Mappings
Second order differential equations
Second order differential equations - Systems of Linear Differential Equations
Second order differential equations - Constant-Coefficient Linear Equations & Applications
Second order differential equations - Variable-Coefficient Equations: Substitution Methods
Second order differential equations - Series Solutions and Special Functions
Second order differential equations - Non-Linear, Functional, and Theoretical Problems
Discrete Random Variables
Poisson Distribution
Approximating the Poisson to the Normal distribution
Approximating the Binomial to the Poisson distribution
Probability Generating Functions
Cumulative distribution functions
Exponential Distribution
Bivariate data
Linear regression
Moment generating functions
Linear combinations of normal random variables
Central limit theorem
Unbiased Estimators
Hypothesis test of a normal distribution
Hypothesis test of Pearson’s product-moment correlation coefficient
Hypothesis test of Spearman’s rank correlation coefficien
Hypothesis test of a Poisson distribution
The Gamma Distribution
Chi-squared distribution
Yates’ continuity correction
Non-parametric tests
Wilcoxon tests
Moments of inertia
Dynamics
Rocket Equation
Multivariate Calculus
Unknown Mech
Generating Functions
Functional Equations
Circuits
Stability of Equilibria
Irrelevant
Misc. Stats
Rings & Further Algebra
Markov Chains
Question needs checking before classifying
Conics (off spec)
Geometry (off spec)
Mathematical Essays
Other Essay
Worksheet Citation (for copying)
Click the copy button or select the text to copy this citation for use in worksheets.
Problem Text
Show that the condition that the two triangles in the Argand plane formed by the two triples of complex numbers $a_1$, $a_2$, $a_3$ and $b_1$, $b_2$, $b_3$ should be similar in the same sense is that \[\frac{a_1 - a_3}{a_1 - a_2} = \frac{b_1 - b_3}{b_1 - b_2}.\] The three triangles $BCA'$, $CAB'$, $ABC'$ are similar in the same sense (although they are not necessarily similar to $ABC$). Show that the triangles $ABC$, $A'B'C'$ have the same centroid.
Solution (Optional)
None
Preview
Problem
Solution
Update Problem
Cancel
Current Ratings
Difficulty Rating:
1500.0
Difficulty Comparisons:
None
Banger Rating:
1485.5
Banger Comparisons:
1
Search Problems
Press Enter to search, Escape to close