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 |