Circle (A₂): Bankoff's Triplet circle

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

Definition

Let (O₃) be the incircle of the arbelos. The circle through C and through the points of tangency of (O₃) with (O₁) and (O₂) respectively is Archimedean. [Bankoff 1954]

Properties

• (A₂) is the incircle of triangle O₁O₂O₃.
• (A₂) is tangent to (O'), the (semi)circle on diameter (O₁O₂), the point of tangecy lies on the line connecting C with the point of tangency of (O₃) and (O). [van Lamoen 2006]
• Two points of (A₂) can also be found by intersecting the line M₁M₂ connecting the two midpoints of the semicircular arcs (O₁) and (O₂). This line intersects CD in a second point of (A₂) (the first is C) and intersects (O') in its highest point and in the point of tangency T of (O') and (A₂).
  
• The center A₂ can be found as the intersection of O₂M₁ and O₁M₂. [Yiu 1998] See below for a stronger statement.
• The point T can also be found as the intersection of (AM₁O₂) and (BM₂O₁). These circles also hit the common points of (O₂) and (O₃) and of (O₁) and (O₃) respectively. [van Lamoen 2006]
  
• The point T lies on the line connecting C and the point of tangency of (O₃) and (O). [van Lamoen 2006]
• The point M₁ is the external center of similitude of (A₂) and (O₂), while M₂ is the external center of similitude of (A₂) and (O₁). Also, let M' be the midpoint of the arc BA not bordering the arbelos. Then M' is the external center of similitude of (A₂) and (O₃) [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 (A₂) and CD. [Bui 2007]
  
• Let M be the highest point of (O) and L the lowest point of the incircle (O₃). Then let L' be the orthogonal projection of L on AB. Them A₂ is the intersection of  L'M and OL. [FvL, 20 Sep 2007]
  
• (A₂) is the incircle of the square inscribed in triangle ABO₃ with one side of the square on AB. [FvL, 3 Mar 2008]
  

Generalizations

• The circles through C and through the points of tangency with (O₁), respectively (O₂), of the nth circle in the Pappus chain tangent to (O) and (O₁), respectively (O₂) - where (O₃) 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 (O₁), respectively (O₂), of the nth circle in the Pappus chain tangent to (O₁) and (O₂) - where (O₃) is the first circle of this chain, have radius 1/n times the Archimedean circle (are 1/n-Archimedean) [Danneels and van Lamoen 2007]
  

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