| Dec 29 |
Three Circles and Area |
| Dec 29 |
Is X a Midpoint of a Chord |
| Dec 27 |
Proportions and the Incenter |
| Dec 25 |
Limit of a Recursive Sequence |
| Dec 23 |
An Inequality with Constraint VI |
| Dec 21 |
Pythagorean Perimeter Theorem |
| Dec 21 |
Generalized Carnot's theorem |
| Dec 19 |
Inequality with Csc And Sin |
| Dec 19 |
Euler Inequality in Triangle |
| Dec 19 |
Concurrence on Angle Bisector |
| Dec 18 |
An Inequality with Constraint V |
| Dec 18 |
Jensen's Inequality |
| Dec 17 |
An Inequality with Constraint IV |
| Dec 16 |
An Inequality from Morocco |
| Dec 16 |
An Inequality from Morocco, with a Proof, or Is It? |
| Dec 15 |
A Limit of Another Sequence |
| Dec 14 |
A Limit of a Sequence |
| Dec 13 |
Optimization in Parallelepiped |
| Dec 12 |
Dao's Variant of Thebault's First Problem |
| Dec 11 |
Jordan Curve Theorem |
| Dec 9 |
An Inequality with Constraint III |
| Dec 8 |
Solution #10 to a variant of the 80-80-20 triangle problem |
| Dec 8 |
Solution #9 to a variant of the 80-80-20 triangle problem |
| Dec 8 |
Solution #8 to a variant of the 80-80-20 triangle problem |
| Dec 7 |
A Problem with Two Isosceles Triangles |
| Dec 4 |
Advancing a Millenium Problem |
| Dec 4 |
An Inequality with Constraints II |
| Dec 3 |
An Inequality with Constraint |
| Nov 30 |
Review of Problems for Metagrobologists by David Singmaster |
| Nov 29 |
Steiner's Circumellipse in Barycentric Coordinates |
| Nov 28 |
Steiner's Inellipse in Barycentric Coordinates |
| Nov 27 |
Area of the Intersection of Two Equal Rectangles |
| Nov 26 |
Pedal Parallelogram |
| Nov 26 |
Cevian Parallelogram |
| Nov 24 |
Constrained Area in Triangle |
| Nov 24 |
Area of Cutoff Triangles |
| Nov 24 |
Area of Cevian Triangle |
| Nov 24 |
Area Inequalities in Triangle |
| Nov 22 |
Inequality from Math Phenomenon |
| Nov 22 |
Chebyshev Inequality |
| Nov 21 |
Rearrangement Inequality |
| Nov 19 |
Sanchez's Viviani's Area Analogue |
| Nov 19 |
Equal areas In Circle |
| Nov 18 |
A Problem with Right Isosceles Triangles |
| Nov 17 |
Bicubal Domino |
| Nov 16 |
Constrained Intermarriages |
| Nov 15 |
Bisectors and Proportions |
| Nov 10 |
Newton's and Maclaurin's Inequalities |
| Nov 9 |
Getting Your Rightful Share Back |
| Nov 6 |
An Inequality from Tibet |
| Nov 4 |
Application of Cauchy-Schwarz Inequality |
| Nov 3 |
Optimal Quadrilateral Inscribed in Square, PWW |
| Nov 2 |
Optimal Quadrilateral Inscribed in Square |
| Oct 29 |
Simple But Uncommon Equation from 1968 IMO |
| Oct 29 |
A Functional Equation from 1968 IMO |
| Oct 27 |
Inequalities and Quadratic Forms |
| Oct 27 |
An Application of Schur's Inequality |
| Oct 26 |
Schur's Inequality |
| Oct 25 |
An Inequality in Determinants |
| Oct 23 |
Fold Square into Equilateral Triangle |
| Oct 22 |
Areas in Three Squares |
| Oct 21 |
Orthogonality in Two Squares |
| Oct 19 |
An Equation in Matrices |
| Oct 19 |
The Shortest Crease |
| Oct 17 |
An Equation in Integers and Rational Numbers |
| Oct 16 |
Pythagorean Theorem, Proof 114 |
| Oct 16 |
Pythagorean Theorem by Inscribed Semicircle |
| Oct 15 |
Golden Ratio in Isosceles Right Triangle, Square, and Semicircle |
| Oct 14 |
Trisect Segment: 2 Circles, 4 Lines |
| Oct 13 |
Extremal Problem in a Quadrilateral |
| Oct 11 |
Centroids and Circumcenters |
| Oct 9 |
Butterfly in Inscriptible Quadrilateral |
| Oct 8 |
Prove 2[sqrt(n) - sqrt(n-1)] > 1/sqrt(n) |
| Oct 7 |
Angles in a Triangle: an Exercise |
| Oct 2 |
Thales on Angle Bisectors |
| Sep 25 |
Angle Chasing on Radical Axis |
| Sep 24 |
Angle in Square |
| Sep 21 |
An Application of Hlawka's Inequality |
| Sep 19 |
Scalar Product Optimization |
| Sep 17 |
Tangents And Bisectors |
| Sep 14 |
Equation in Permutations |
| Sep 12 |
Triangle by HM segments |
| Sep 9 |
Transversal Trigonometry |
| Sep 9 |
A Property of Cyclic Quadrilateral |
| Sep 8 |
Miguel Ochoa's van Schooten is a Slanted Viviani |
| Sep 8 |
Miguel Ochoa's van Schooten Like Theorem III |
| Sep 7 |
Miguel Ochoa's van Schooten Like Theorem II |
| Sep 6 |
On the Intersection of 1+x and e^x and Trigonometric Consequences |
| Sep 4 |
A Problem of Acute Triangle |
| Sep 3 |
When a Triangle is A-cute |
| Sep 1 |
Homothety in Equilateral Triangles |
| Aug 31 |
What 1, 6, and 14 Have to Do with 5 |
| Aug 30 |
Concurrence in Equilateral Triangle |
| Aug 28 |
Inequality in Rectangle |
| Aug 27 |
The Dual of Maxwell's Theorem |
| Aug 25 |
Area of Trapezoid |
| Aug 24 |
Extremal Distance Ratio |
| Aug 23 |
An Identity in Triangle II |
| Aug 5 |
Algebraic-Geometric Equation |
| Aug 4 |
Modern Day Sangaku |
| Aug 2 |
Amazing Mix of Triangular Numbers and Golden Ratio |
| Jul 28 |
An Unusual Quadratic Equation |
| Jul 27 |
Regular Hexagon from Arbitrary Triangle |
| Jul 26 |
Inequality for Integer Sequences II |
| Jul 26 |
Inequality in Acute Triangle |
| Jul 26 |
Inequality for Integer Sequences |
| Jul 25 |
Perspective Triangles in Circle |
| Jul 24 |
Relation Engendered By Medians |
| Jul 24 |
Relation Engendered By Altitudes |
| Jul 23 |
Cubic Equation for the Sides of a Triangle |
| Jul 23 |
Property of Quadratic Polynomials |
| Jul 22 |
Pleasant Proportions in Triangle |
| Jul 21 |
An Inequality via Young's Theorem |
| Jul 21 |
An Angle Inequality in Simple Polygons |
| Jul 20 |
Reuleaux's Triangle, Extended |
| Jul 14 |
Triangle from Two Side Lines and the Euler Line |
| Jul 10 |
An Universal Inequality for Cevians |
| Jul 1 |
Iterations on Monotone Functions |
| Jun 30 |
Tangential Chaos |
| Jun 29 |
An innovative recurrence |
| Jun 25 |
An Equation in Determinants |
| Jun 24 |
Two Related Triangles of Equal Areas |
| Jun 22 |
Leo's Lemma, Second Application |
| Jun 21 |
Two Sequences Defined by Recurrence |
| Jun 19 |
Three visions of a fact |
| Jun 17 |
Simpson Paradox
|
| Jun 17 |
Importance of Having an Angle of 60 Degrees |
| Jun 17 |
Triangle with a 60 degrees angle |
| Jun 16 |
A Sum of a Double Series |
| Jun 16 |
Leo Giugiuc's Trigonometric Lemma |
| Jun 15 |
Is There a Limit |
| Jun 14 |
A One-Line Proof of the Infinitude of Primes |
| Jun 11 |
Clough's theorem: the simplest proof |
| Jun 10 |
Beyond Viviani |
| Jun 10 |
Slanted Viviani, PWW |
| Jun 8 |
Area Inequality in Triangle II |
| Jun 7 |
Slanted Viviani |
| Jun 6 |
YATPOPT, Yet Another Trigonometric Proof of the Pythagorean Theorem |
| Jun 5 |
Problem in Direct Similarity |
| Jun 4 |
Inverse Similarity through Reflections |
| Jun 2 |
An Inequality with Nested Radicals |
| May 31 |
From Triangle Inequality to Inequality in Triangle |
| May 28 |
A Curious Identity in Triangle |
| May 26 |
An Inequality in Triangle, II |
| May 26 |
The Jeweler's Observation, a look back |
| May 25 |
Isotomic Reciprocity |
| May 20 |
Construct Triangle from Altitude, Median and Inradius |
| May 18 |
An Affine Property of Barycenter |
| May 18 |
An Identity for Regular Tetrahedra |
| May 16 |
Van Khea's Fourth Identity in Triangle |
| May 12 |
Parabolas Related to the Orthic Triangle |
| May 11 |
An Inequality in Triangle |
| May 11 |
Parabolic Reciprocity |
| May 10 |
Miguel Ochoa's van Schooten Like Theorem |
| May 10 |
Elegant Minimum of a Function |
| May 7 |
One More Seven Circles Theorem |
| May 7 |
William Wallace's 1803 Statement of the Butterfly Theorem |
| May 6 |
Third Unusual Identity in Triangle |
| May 4 |
Dan-Stefan Marinescu Inequality |
| May 4 |
Twin Segments in Four Semicircles |
| May 2 |
Characterization of Trapezoid |
| May 1 |
A wrapping surprise |
| Apr 30 |
Review of The Proof and The Pudding by Jim Henle |
| Apr 28 |
From Arbitrary Pentagon to Regular One |
| Apr 25 |
A Geometric Proof of Proizvolov's Identity |
| Apr 24 |
From Arbitrary Hexagon to Regular One |
| Apr 22 |
Extras in Bottema's Configuration |
| Apr 22 |
From Straight Line to Regular Hexagon |
| Apr 21 |
Parallelogram in Parallelogram And in Projection |
| Apr 20 |
A Problem in Three Pentagons |
| Apr 19 |
A Generalization of Simson Line |
| Apr 17 |
A Combinaton Locus |
| Apr 16 |
Golden Ratio in Mixtilinear Circles |
| Apr 16 |
Proofs of the Pythagorean Theorem via Heron's Formula II |
| Apr 15 |
Further Properties of Peculiar Circles |
| Apr 14 |
An Inequality from the 2015 Romanian TST |
| Apr 6 |
Another Identity in Triangle |
| Apr 5 |
Circle of Apollonius in Equilateral Triangle |
| Apr 5 |
Integer Triangle That Sums to Cube |
| Apr 5 |
Another Sangaku in Square |
| Apr 3 |
An Identity in Triangle |
| Apr 3 |
Square Roots and Triangle Inequality |
| Apr 1 |
Why Learn Mental Math Tricks? |
| Mar 30 |
Four Golden Circles |
| Mar 27 |
Construct Triangle by Angle, Median, and Circumradius |
| Mar 27 |
Two Conics, Pascal, Chasles, and Cross-Ratio |
| Mar 26 |
Construct Triangle by Angle, Altitude and Median |
| Mar 24 |
A Conic in a Triangle |
| Mar 22 |
Concurrence in Arbelos |
| Mar 21 |
Areas in Two Parallelograms and a Translation |
| Mar 19 |
Triangle from Angle Bisector, Altitude and Inradius |
| Mar 19 |
Triangle from Median, Altitude and Circumradius |
| Mar 18 |
An Equation Involving the Total Number of Digits |
| Mar 16 |
Triangle from Angle Bisector, Altitude and Circumradius |
| Mar 16 |
An Equation Involving Four Collinear Points |
| Mar 16 |
Hlawka's Inequality |
| Mar 13 |
Area Inequality in Triangle |
| Mar 11 |
Construct Triangle from Angle, Inradius, and Altitude |
| Mar 11 |
Triangle from Angle, Inradius, and Difference of Sides |
| Mar 10 |
Infinitude of Primes As a Quickie |
| Mar 5 |
Between a Cross and a Square |
| Mar 4 |
Infinitude of Primes Via Powers of 2 |
| Mar 4 |
Sophie Germain's Identity |
| Mar 3 |
A Property of Number 4 |
| Mar 2 |
Equilateral Triangle on a Closed Curve |
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Copyright © 1996-2018 Alexander Bogomolny
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