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

The Cross-Touch Perspector

On Novermber, 2013, I discovered this point and inmediately sent it to Clark Kimberling and Peter Moses. The point is not listed on ETC, but, seemingly, has not special properties.

The configuration consist in a triangle with three semi-circles on  each side and three common tangents lines. Now from each vertice draw a line joining the farthest tangenticial point on its corresponding semicircle and cyclic. The three intersecting points form a triangle wich is in perspective with the reference triangle.

I sometimes donate my results to AoPS when they have no special interest out of problem-solving, so I did. Next day, someone under the nick "Lyub4o" posted a proof which you can see here.

The internal version is the incenter of the reference triangle. I will take the liberty to name the external point "The Cross-Touch Perspector".