# 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]
• Let P and Q on (O1) and (O2) respectively be such that PO1 and QO2 are parallel. Let PO2 and QO1 intersect in A2gen. The circle with center A2gen through O is Archimedean [Phil Todd, Oct 2008]

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