A study of form, relation, and proof

ELEMENTARY
GEOMETRYBeauty in Problems, Proofs, and Diagrams

A growing collection of Euclidean geometry: carefully stated problems, closely observed configurations, and precise proofs and diagrams. The emphasis is on seeing why a configuration works.

Featured classical note

The Euler Line from the Medial Triangle

A homothety centered at the centroid

The circumcenter of a triangle is the orthocenter of its medial triangle, and a centroid-centered homothety carries the circumcenter to the original orthocenter.

Read the note Classical Geometry

Triangle ABC with medial triangle DEF; the three altitudes of DEF meet at O, the circumcenter of ABC. — The circumcenter O of ABC is the orthocenter of the medial triangle DEF.

Selected solution

A Solution to Crux Mathematicorum Problem 5021

Crux Mathematicorum Problem 5021, followed by a direct-similarity generalization proved with Pappus's theorem and a radical axis.

Projective geometry · Circles · Transformations

Read the problem and solution

Two directly similar triangles OAB and OCD. Lines AE and DF meet at P, while AC and BD meet at Q. The points O, Q, and P are collinear. The two circumcircles have centers K and L, and OP is perpendicular to KL. — The Pappus configuration and two circumcircles used in the generalized solution.

Methods: supporting formulas and their meaning

More problems

Further problems and solutions

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Solution

A Solution to Crux Mathematicorum Problem 4941

Crux Mathematicorum Problem 4941, followed by an incircle-inversion proof using the medial triangle of the contact triangle.

16 August 2026Read

Solution

A Solution to Crux Mathematicorum Problem 5031

Crux Mathematicorum Problem 5031, followed by an incircle-inversion proof using contact-chord midpoints and a cyclic quadrilateral.

10 August 2026Read

Solution

A Solution to Crux Mathematicorum Problem 5088

Crux Mathematicorum Problem 5088, followed by an angle-chasing proof through tangency and cyclic quadrilaterals.

10 August 2026Read

Index

Main topics

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Algebra5 classical · 1 method · 1 solution Circles6 classical · 2 methods · 6 solutions Conics1 classical · 1 solution Isogonal conjugation1 classical Projective geometry2 solutions Transformations4 classical · 4 solutions Triangle centers8 classical Triangle geometry10 classical · 2 methods · 1 original problem · 6 solutions