Three Parabolas with Common Directrix

Description and proof

Seven Problems in Equilateral Triangle

Description and proofs

Circle Rotations and Fixed Points

Description and proof

Extra eyeballing leads to an Archimedean triplet!

Description and proof

Garcia's Archimedean Quadruplets

One of the properties of the arbelos noticed and proved by Archimedes in his Book of Lemmas is that the two small circles inscribed into two pieces of the arbelos cut off by the line perpendicular to the base through the common point of the two small semicircles are equal. The circles have been known as Archimedes' Twin Circles. More than 2200 years after Archimedes,L. Bankoff (1974) has found another circle equal to the twins. In 1999 a large number of additional circles of the same radius has been reported by Dodge et al. More recently,F. Power described another quadruplet of circles that should be adopted into the family (Taken from CTK). In this page I am adding more circles.

For a description, please, go to Cut-The-Knot.org


Related topics:

Arbelos

Archimedes' Twin Circles

Archimedes' s Quadruplets

Bankoff Circle

Schoch Circles

Schoch Line

A Dozen More Arbelos Twins

Thomas Schoch - My Arbelos Story

Woo Circles

Online Catalogue of Archimedean Circles

Reductio ad absurdum beautifuly used

Theorem:
Not all problems can be solved.

Proof:
Suppose all problems can be solved (1).
We consider following problem P:
“To find one problem which can not be solved”
Because all problems can be solved therefore we can not find this problem. It means problem P can not be solved. It contradict (1) therefore “not all problems can be solved”.

The theorem is proved.

By Quang Tuan Bui

Source

Shifted by the Inradius Tangent Collinearities

Description and proof