SixPivots

The twelve pivots of a triangle (DEF) inscribed in another triangle (ABC)

There are actually 12 pivots about which you can [turn] a triangle inscribed in another triangle. Here by inscribed we mean a triangle similar to DEF having each one of its vertices on a (different) side of ABC. The pivots lie by six on two circles which are inverse to each other w.r. to the circumcircle of the triangle. By this inversion pivots on the two circles are paired. Below are shown six pivots and the corresponding six triangles (all similar to each other) inscribed with respect to these pivots, which lie on a circle. Look at the file Pivot.html for the definition and the meaning of the pivots.

The two circles containing the 12 pivots belong to the family generated by the Lemoine axis and the circumcircle. Hence their centers are on the Brocard axis. In addition the six points on each circle are, by three, orbits of the Moebius recycler of the triangle ABC (the orbits are correspondingly the vertices of triangles P₁P₃P₅ and P₂P₄P₆). For a review of the definition and elementary properties of the Moebius recycler look at the file Recycler.html .
Note that the six pivots pictured above are created from P₁ by applying repeatedly inversions with respect to the Apollonian circles of the triangle. Note that the six sides of triangles P₁P₃P₅ and P₂P₄P₆ are tangent to a conic (e).

The twelve pivots of a triangle (DEF) inscribed in another triangle (ABC)

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