New construction of the Tarry point
- Hello all in geometry,
I have always found the Tarry point mysterious. It is opposite the
Steiner point in the circumcircle and the intersection of the
circumcircle with the Kiepert hyperbola. Both of these are nice, if
I have a new construction which both explains the above and shows
that the Tarry point is one example of a universal phenomenon.
We begin as I always begin, with a single point in the plane of the
triangle. We will choose this point to be K, the symmedian point.
There are now two natural lines, the dual of K and G—K. Natural in
this context means that the structure is affine invariant. The
endpoints of these lines map to antipodal points on the Steiner ellipse.
These lines meet at P. The isotomic of the two lines are
circumconics, one, the Kiepert hyperbola, has its perspector tS at
infinity, the second has perspector K. The isotomic of P is the
intersection between these conics. This is the Tarry point.
The constuction shows that this is construction is general. Slightly
more analysis shows that S and T are opposite on the K circumconic
and that this too is general.
Triangle web page: