With this third congruent circle Leon Bankoff opened the quest for Archimedean circles in 1954.

Let (O_{3}) be the incircle of the arbelos. The circle through C and
through the points of tangency of (O_{3}) with (O_{1}) and (O_{2})
respectively is Archimedean. [Bankoff 1954]

- (A
_{2}) is the incircle of triangle O_{1}O_{2}O_{3}. - (A
_{2}) is tangent to (O'), the (semi)circle on diameter (O_{1}O_{2}), the point of tangecy lies on the line connecting C with the point of tangency of (O_{3}) and (O). [van Lamoen 2006] - Two points of (A
_{2}) can also be found by intersecting the line M_{1}M_{2}connecting the two midpoints of the semicircular arcs (O_{1}) and (O_{2}). This line intersects CD in a second point of (A_{2}) (the first is C) and intersects (O') in its highest point and in the point of tangency T of (O') and (A_{2}).

- The center A
_{2}can be found as the intersection of O_{2}M_{1}and O_{1}M_{2}. [Yiu 1998]*See below for a stronger statement.* - The point T can also be found as the intersection of (AM
_{1}O_{2}) and (BM_{2}O_{1}). These circles also hit the common points of (O_{2}) and (O_{3}) and of (O_{1}) and (O_{3}) respectively. [van Lamoen 2006] - The point T lies on the line connecting C and the point of tangency of
(O
_{3}) and (O). [van Lamoen 2006] - The point M
_{1}is the external center of similitude of (A_{2}) and (O_{2}), while M_{2}is the external center of similitude of (A_{2}) and (O_{1}). Also, let M' be the midpoint of the arc BA not bordering the arbelos. Then M' is the external center of similitude of (A_{2}) and (O_{3}) [FvL, 24 Oct 2006].

- Let T be the point on CD beyond D such that CT=AB. Then the orthocenter
H of triangle ABT is the second intersection of (A
_{2}) and CD. [Bui 2007] - Let M be the highest point of (O) and L the lowest point of the incircle
(O
_{3}). Then let L' be the orthogonal projection of L on AB. Them A_{2}is the intersection of L'M and OL. [FvL, 20 Sep 2007]

- (A
_{2}) is the incircle of the square inscribed in triangle ABO_{3}with one side of the square on AB. [FvL, 3 Mar 2008]

- The circles through C and through the points of tangency with (O
_{1}), respectively (O_{2}), of the nth circle in the Pappus chain tangent to (O) and (O_{1}), respectively (O_{2}) - where (O_{3}) is the first circle of this chain, have radius n times the Archimedean circle (are n-Archimedean) [Danneels and van Lamoen 2007]

- The circles through C and through the points of tangency with (O
_{1}), respectively (O_{2}), of the nth circle in the Pappus chain tangent to (O_{1}) and (O_{2}) - where (O_{3}) is the first circle of this chain, have radius 1/n times the Archimedean circle (are 1/n-Archimedean) [Danneels and van Lamoen 2007]

- Let P and Q on (O
_{1}) and (O_{2}) respectively be such that PO_{1}and QO_{2}are parallel. Let PO_{2}and QO_{1}intersect in A_{2gen}. The circle with center A_{2gen}through O is Archimedean [Phil Todd, Oct 2008]