| Dec 30 |
Pedal Triangles of Inverse Points |
| Dec 28 |
Proclivity of Conjoint Archimedean Twins for Proliferation |
| Dec 26 |
A Pedal Property in Equilateral Triangle |
| Dec 26 |
Pedal Similarity |
| Dec 25 |
Sides and Area of Pedal Triangle |
| Dec 24 |
Extension of Euclid's Game |
| Dec 23 |
Parallelogram with Side Lines through Fixed Points |
| Dec 22 |
BMO 2014, Problem 3 |
| Dec 21 |
An Inequality Involving Fermat Point |
| Dec 20 |
Inequalities in Triangle |
| Dec 18 |
Bounds for the Sum of Distances to the Vertices |
| Dec 17 |
Garcia-Feuerbach Collinearity |
| Dec 15 |
Archimedean Siblings out of Wedlock, i.e., Arbelos |
| Dec 6 |
Tangent to the Incircle |
| Dec 4 |
Sum of All Natural Numbers |
| Dec 3 |
Dao Thanh Oai's Invariant |
| Dec 3 |
Chain of Six Intersecting Circles |
| Dec 3 |
Chain of Four Intersecting Circles |
| Dec 2 |
Review of The Best Writing on Mathematics 2014 |
| Nov 24 |
Further properties of Van Aubel Configuration |
| Nov 22 |
Extra Feature of Van Aubel Configuration |
| Nov 21 |
Triangle from angle A, m_b, h_a |
| Nov 21 |
Law of Cosines by Ancient Sliding |
| Nov 20 |
Midpoint Reciprocity in Napoleon's Configuration |
| Nov 19 |
Longitude, Latitude and Distance to the Equator |
| Nov 18 |
A Problem for the 2015 New Year's Eve |
| Nov 17 |
Property of Two Pencils of Parallel Lines |
| Nov 16 |
Triangle by Inradius, Circumradius, Side or Angle |
| Nov 14 |
Train on Bridge |
| Nov 12 |
Uncommon Triangle Constructions |
| Nov 6 |
Equilateral Triangles On Sides of a Parallelogram II |
| Nov 4 |
Fermat Point Several Times Over |
| Nov 3 |
Van Schooten's and Pompeiu's Theorems |
| Nov 2 |
The Law of Cosines - Yet Another PWW |
| Nov 1 |
Triangle from Side, Median, and Altitude |
| Oct 31 |
Mathematicians and Musicians |
| Oct 30 |
Tables for Multiplication |
| Oct 29 |
Using Math Rules: An Example |
| Oct 28 |
Broken Line in Triangle |
| Oct 27 |
Cutting Circle to Rearrange Two Dots |
| Oct 27 |
Two Jugs - One's Volume Unknown |
| Oct 23 |
Triangle from Side, Circumradius, and Median |
| Oct 21 |
Vietnamese Extension of a Japanese Theorem |
| Oct 21 |
Chasing Angles Among Angle Bisectors |
| Oct 19 |
Vietnamese Lemma towards an Extension of a Japanese Theorem |
| Oct 17 |
Eyeball theorem |
| Oct 16 |
Construct Triangle by Angle Bisector, Altitude, and Side |
| Oct 14 |
Properties of the Harmonic Ratio |
| Oct 13 |
A. G. Samosvat's Proof by Dissection of the Pythagorean Theorem (#104) |
| Oct 10 |
Two Circles in Square II |
| Oct 9 |
(Un)fair Division by Consensus |
| Oct 9 |
Circle Concurrence on Circumcircle |
| Oct 9 |
Isogonal Lemma |
| Oct 7 |
Review of 'Zombies and Calculus' by Colin Adams |
| Oct 3 |
Dao's Archimedean Twins - Third Pair |
| Oct 2 |
Dao's Archimedean Twins - Second Pair |
| Oct 1 |
Beauty and the Beast - in Trigonometry |
| Oct 1 |
Gladiator Game |
| Sep 30 |
Twins Beget Twins |
| Sep 29 |
Construction of Triangle from Side, Angle, and that Angle's Bisector |
| Sep 24 |
Projective Invariants For Pascal And Desargues |
| Sep 19 |
Deceptive Appearances |
| Sep 18 |
Another Seven Circles Theorem, now with proof |
| Sep 17 |
Morley's Equilaterals, Spiridon A. Kuruklis' proof |
| Sep 15 |
Backward Proof of Morley's Theorem by B. Stonebridge |
| Sep 14 |
Review of Coffee, Love and Matrix Algebra |
| Sep 12 |
Powers of 6 and 7771 |
| Sep 11 |
With One Weighing |
| Sep 11 |
Solutions of F(x)=x and F(F(x))=x |
| Sep 8 |
Review of Really Big Numbers |
| Sep 7 |
Dijkstra's Proof of Morley's Theorem |
| Sep 4 |
A Property of Unimodal Integers as a Whole |
| Sep 3 |
Symmetries in Triangle II |
| Sep 1 |
Symmedian in Right Triangle |
| Aug 10 |
Concyclicity from Collinearity |
| Aug 5 |
Moment of Inertia of a Regular Polygon |
| Aug 4 |
Novel Concurrency of Perpendicular Bisectors |
| Aug 4 |
Garcia's Two Circles Lemmas |
| Aug 1 |
Distance to the Horizon on the Fourth of July |
| Jul 29 |
Butterflies in a Pencil of Conics |
| Jul 28 |
Theorem of Three Tangents to a Conic |
| Jul 25 |
Projections of Convex Quadrilateral |
| Jul 23 |
A New Proof That the Rationals Are Countable |
| Jul 22 |
Three Parabolas with Common Directrix |
| Jul 19 |
The 80-80-20 Triangle Problem, 20-30 case |
| Jul 19 |
The 80-80-20 Triangle Problem, Solution #12 |
| Jul 17 |
Equilateral Triangle on Three Lines |
| Jul 16 |
There is no Difference Between Equilateral Triangles |
| Jul 15 |
Equilateral Triangle on Angle Bisectors |
| Jul 10 |
Concyclic Points at the IMO 2014 |
| Jul 10 |
Butterfly in Parabola |
| Jul 10 |
Reflection in a Triangle Tangent to Parabola |
| Jul 9 |
Eyeballs Projected |
| Jul 8 |
An Increasing Sequence from 2014 IMO |
| Jul 8 |
Multiplication of Points on an Ellipse |
| Jul 6 |
Multiplication of Points on a Circle |
| Jul 4 |
Shedding Light on the Ball for Eyeballing |
| Jul 3 |
Invert Two Circles Into Equal Ones |
| Jul 2 |
Seven and the Eighth Circle Theorem |
| Jul 1 |
Two Triangles Sharing the Circumcenter |
| Jul 1 |
Gergonne's Apollonian Circles |
| Jun 30 |
Golden Ratio in Regular Pentagon |
| Jun 28 |
Line Through a Center of Similarity |
| Jun 28 |
Line Not Through a Center of Similarity |
| Jun 25 |
Six Concyclic Points on Sides of a Triangle |
| Jun 23 |
Bespectacled Eyeballs Extension |
| Jun 23 |
Segment Projections in Equilateral Triangle |
| Jun 23 |
A Property of Perspective Triangles |
| Jun 20 |
Rectified, Halved, Sheared, Eyeballs Still Surprise |
| Jun 20 |
Symmetry in an Asymmetric Configuration |
| Jun 19 |
Eyeball Theorem Rectified |
| Jun 18 |
Dao's Archimedean Twins |
| Jun 17 |
Tangent Circles, Similarity and Equal Segments |
| Jun 16 |
Updated Garcia's Archimedean Twins |
| Jun 16 |
Updated the Praying Eyes Theorem |
| Jun 15 |
Garcia's Archimedean Twins |
| Jun 15 |
Eyeballing a ball |
| Jun 14 |
Lines Crossing Circles at Vertices of Equilateral Triangle |
| Jun 13 |
Square, Semicircle, and Other Curves |
| Jun 12 |
A Property of Points on Excircle |
| Jun 12 |
Similar Triangles in Crossing Circles |
| Jun 12 |
Semicircle on Square |
| Jun 11 |
Sequences of Concyclic Points on a Conic |
| Jun 9 |
Lines through Circles at Vertices of Equilateral Triangle |
| Jun 9 |
Circles Tangent to Medians at the Centroid |
| Jun 7 |
Shifted by the Inradius Tangent Collinearities |
| Jun 3 |
A Property of Points on Incircle |
| Jun 2 |
Tangent Curiosity in Equilateral Triangle |
| Jun 1 |
Pythagoras, with a Nod to the Carpets Theorem |
| May 30 |
An Equation in Squares |
| May 28 |
Bottema in Three Rotations |
| May 27 |
Bottema with Similar Triangles |
| May 27 |
Bottema with a Product of Rotations |
| May 25 |
Concurrent and Parallel Lines in Parallelogram |
| May 25 |
Looking Back at Bottema |
| May 24 |
Parallelogram and Four Equilateral Triangles |
| May 23 |
A Series with k-Permutations |
| May 22 |
Regular Hexagon and Triangle Joined at Vertex |
| May 22 |
Yet Another Generalization of Bottema's Theorem |
| May 22 |
A DIY Proof of the Irrationality of sqrt(2) |
| May 21 |
Irrationality of k-th roots |
| May 20 |
Entry 1.414... in Ramanujan's Notebooks |
| May 18 |
Carpets in Triangle, II |
| May 18 |
Irrationality of sqrt(2) via a Difference Equation |
| May 17 |
An Inequality with Permutations, II |
| May 15 |
An Inequality with Permutations, I |
| May 14 |
Spiral Similarity Leads to Equilateral Triangle |
| May 13 |
A Matter of Collinearity: Metamorphosis of a Problem |
| May 12 |
Four Incircles in an Equilateral Triangle, Third Solution |
| May 12 |
Reflection in the Circumcenter |
| May 11 |
Another Algebraic Proof of the Pythagorean Theorem |
| May 6 |
Two Pencils of Parallel Lines |
| May 4 |
Telescoping Squares |
| May 3 |
Telescoping Cubes |
| May 2 |
An Olympiad Problem With the Floor Function |
| May 2 |
A Number Theoretic Problem With the Floor Function |
| Apr 30 |
Pythagorean Triples via Fibonacci Numbers |
| Apr 29 |
Viviani's 3D Analogue |
| Apr 29 |
Tiling Sphere by Four Equal Triangles |
| Apr 28 |
Tetrahedron with Orthogonal Perpendiculars to Opposite Edges |
| Apr 28 |
Tetrahedron with Equiareal Faces, II |
| Apr 28 |
Isosceles Tetrahedron |
| Apr 27 |
Tetrahedron with Equiareal Faces |
| Apr 26 |
An Algebraic Lemma with Geometric Consequences |
| Apr 24 |
Tetrahedron in Parallelepiped |
| Apr 23 |
Review of Count Like an Egyptian |
| Apr 22 |
Fixed Point on Perpendicular Bisector |
| Apr 21 |
Sum of Absolute Values |
| Apr 19 |
A Quartic Equation |
| Apr 18 |
ln(2) in Pascal's Triangle |
| Apr 17 |
pi^e vs e^pi |
| Apr 17 |
Pi in Pascal's Triangle via Triangular Numbers |
| Apr 16 |
Hjelmslev's Theorem on Canvas |
| Apr 14 |
Pi in Pascal's Triangle |
| Apr 13 |
L. Giugiuc's Proof of Morley's Theorem |
| Apr 12 |
B. Bollobás' Trigonometric Proof of Morley's Theorem |
| Apr 9 |
Review of Math Bytes by Tim Chartier |
| Apr 9 |
Cubes in Pascal's Triangle |
| Apr 8 |
Sylvester's Problem, Lang's solution |
| Apr 8 |
Sylvester's Problem, Steinberg's solution |
| Apr 8 |
Sylvester's Problem, Gallai's solution |
| Apr 8 |
Sylvester's Problem, Kelly's solution |
| Apr 7 |
Sylvester's Problem |
| Apr 6 |
Brahmagupta-Fibonacci Identity |
| Apr 3 |
Number of Regions N Lines Divide Plane |
| Apr 3 |
What Is the Extremal Principle? |
| Apr 1 |
Three Euler Lines That Are Four |
| Mar 31 |
A View on the Circle of Apollonius |
| Mar 31 |
What Is Indirect Proof? |
| Mar 30 |
Existence of the Circumcenter, Indirect Proof |
| Mar 27 |
Seven Problems in Equilateral Triangle, Solution to Problems 5, 6, 7 |
| Mar 27 |
Seven Problems in Equilateral Triangle, Solution to Problem 4 |
| Mar 26 |
Seven Problems in Equilateral Triangle, Solution to Problem 3 |
| Mar 26 |
Seven Problems in Equilateral Triangle, Solution to Problem 2 |
| Mar 26 |
Seven Problems in Equilateral Triangle, Solution to Problem 1 |
| Mar 25 |
When Two Segments Intersect |
| Mar 24 |
Voronoi Diagram |
| Mar 23 |
Another Property of Circle Through the Incenter |
| Mar 21 |
Properties of Circle Through the Incenter |
| Mar 21 |
A Property of Circle Through the Incenter |
| Mar 20 |
Central and Inscribed Angles in Complex Numbers |
| Mar 19 |
Simple Property of Circle Through the Incenter |
| Mar 19 |
Conics in Triangle |
| Mar 19 |
Seven Problems in Equilateral Triangle |
| Mar 18 |
What Is Angle Chasing? |
| Mar 18 |
Strategy and Tactics in Problem Solving |
| Mar 17 |
Pure Angle Chasing III |
| Mar 16 |
Pure Angle Chasing II |
| Mar 15 |
Pure Angle Chasing |
| Mar 14 |
Probability of Matching Socks |
| Mar 12 |
Conic in Hexagon |
| Mar 12 |
McDougall's Generalization of Ptolemy's Theorem |
| Mar 11 |
Conic from Parallel Chords |
| Mar 11 |
Concurrence of Ten Nine-Point Circles |
| Mar 10 |
Reim's Similar Coins IV |
| Mar 10 |
Euler-Poncelet Point |
| Mar 10 |
Reim's Similar Coins III |
| Mar 9 |
Reim's Similar Coins II |
| Mar 9 |
Reim's Similar Coins I |
| Mar 6 |
Thales' Theorems |
| Mar 5 |
Probability of Two Integers Being Comprime |
| Mar 3 |
Probability on Discrete Infinite Sets |
| Mar 2 |
Squares in Pascal's Triangle |
| Mar 2 |
Isoperimetric Theorem For Quadrilaterals |
| Mar 2 |
Parallelograms among Quadrilaterals |
| Mar 2 |
Rectangles among Parallelograms |
| Mar 1 |
Carpets in Triangle |
| Feb 28 |
Triangle of Minimum Perimeter |
| Feb 27 |
Sums of Binomial Reciprocals in Pascal's Triangle |
| Feb 26 |
Equilateral Triangle in Equilateral Triangle |
| Feb 20 |
d'Alembert's Misstep |
| Feb 19 |
Pythagorean Theorem for Reciprocals |
| Feb 18 |
Area and Perimeter of Circle |
| Feb 17 |
Beating a Dead Horse: Mathematics Education Reform - Again |
| Feb 17 |
Volume and Area of Torricelli's Trumpet |
| Feb 16 |
Weighted Dice Problem |
| Feb 15 |
The Schwarz Lantern Explained |
| Feb 15 |
Hidden Orthogonality |
| Feb 14 |
A Generalization of the Pythagorean Theorem |
| Feb 14 |
Elementary Introduction into the Concept of Area |
| Feb 14 |
Area of Rectangle |
| Feb 12 |
Trigonometry According to Ptolemy |
| Feb 12 |
A Family of Cyclic Quadrilaterals |
| Feb 11 |
Altitudes have ears, foot, stem, and root |
| Feb 11 |
Another Property of the 9-Point Circle |
| Feb 11 |
Divide Triangle by Lines Parallel to Base |
| Feb 9 |
A Not Symmetric Inequality |
| Feb 9 |
Infinitude of Primes - Golomb's Topological Proof |
| Feb 8 |
Two Cyclic Inequalities |
| Feb 7 |
Practical Relativity |
| Feb 7 |
e and pi Spoof |
| Feb 5 |
Catalan Numbers in Pascal's Triangle |
| Feb 4 |
Infinitely many proofs that there are infinitely many primes |
| Feb 3 |
Projections of Isogonal Conjugate are Concyclic |
| Feb 3 |
A Property of Isogonal Lines |
| Feb 2 |
Another Pair of Archimedean Twins - a Redux |
| Feb 1 |
Hardy's Example of Non-Serious Theorems |
| Jan 31 |
Outline Mathematics: Diophantine Equation I |
| Jan 31 |
Outline Mathematics: Improving on an Escalator |
| Jan 30 |
Counting Permutations with Fixed Points |
| Jan 29 |
Wizards, Aliens, and Starships |
| Jan 29 |
How to write an equation of the union of two sets |
| Jan 27 |
Symmetry Principle in Probability |
| Jan 27 |
An Identity in (Cyclic) Quadrilaterals |
| Jan 26 |
The Law of Cosines by Larry Hoehn |
| Jan 26 |
A Formula for sin(3x) |
| Jan 26 |
A Vector-based Proof of Morley's Trisector Theorem |
| Jan 25 |
Concyclic Incenters in Bicentric Quadrilateral |
| Jan 24 |
Inscriptible Quadrilateral of Triangle Incenters |
| Jan 24 |
Concyclic Points in Inscriptible Quadrilateral |
| Jan 23 |
Two Quadruplets of Concyclic Points |
| Jan 22 |
Checkmate Puzzle |
| Jan 21 |
Random Points on a Segment |
| Jan 20 |
A Geometric Limit |
| Jan 20 |
A Property of Thebault Circles |
| Jan 18 |
Kordemsky's Palindrome Problem |
| Jan 16 |
Anticenter and Orthocenters |
| Jan 14 |
Not So Hidden Homotheties |
| Jan 11 |
A Property of Thebault Circles |
| Jan 10 |
Construction of Thebault Circles |
| Jan 10 |
What is the Color of the Remaining Ball? |
| Jan 9 |
Concurrence on a Circle |
| Jan 9 |
Angle in Right Triangle |
| Jan 9 |
Two Circles, Tangents and Two Collinearities |
| Jan 9 |
Construction and Properties of Mixtilinear Incircles 2 |
| Jan 8 |
Conic in Mixtilinear Incircles |
| Jan 8 |
Construction and Properties of Mixtilinear Incircles |
| Jan 8 |
Collinear Intersections and Products of Ratios |
| Jan 7 |
Yet Another Seven Circles Theorem |
| Jan 6 |
Garfield's Ghost Revisited - by Tony Foster |
| Jan 6 |
Isosceles Trapezoid In Cyclic Quadrilateral |
| Jan 6 |
Droz-Farny Theorem - an Analytic Solution |
| Jan 4 |
A Property of Pascal's Hexagon Pascal May Have Overlooked |
| Jan 2 |
Yet Another Concurrence on the 9-Point Circle |
| Jan 1 |
Six Concyclic Points via Antipedal Triangle |
