Hofstadter points explained

In plane geometry, a Hofstadter point is a special point associated with every plane triangle. In fact there are several Hofstadter points associated with a triangle. All of them are triangle centers. Two of them, the Hofstadter zero-point and Hofstadter one-point, are particularly interesting.^[1] They are two transcendental triangle centers. Hofstadter zero-point is the center designated as X(360) and the Hofstafter one-point is the center denoted as X(359) in Clark Kimberling's Encyclopedia of Triangle Centers. The Hofstadter zero-point was discovered by Douglas Hofstadter in 1992.^[1]

Hofstadter triangles

Let be a given triangle. Let be a positive real constant.

Rotate the line segment about through an angle towards and let be the line containing this line segment. Next rotate the line segment about through an angle towards . Let be the line containing this line segment. Let the lines and intersect at . In a similar way the points and are constructed. The triangle whose vertices are is the Hofstadter -triangle (or, the -Hofstadter triangle) of .^{[2] [1]}

Special case

• The Hofstadter 1/3-triangle of triangle is the first Morley's triangle of . Morley's triangle is always an equilateral triangle.
• The Hofstadter 1/2-triangle is simply the incentre of the triangle.

Trilinear coordinates of the vertices of Hofstadter triangles

The trilinear coordinates of the vertices of the Hofstadter -triangle are given below:

Hofstadter points

For a positive real constant, let be the Hofstadter -triangle of triangle . Then the lines are concurrent.^[3] The point of concurrence is the Hofstdter -point of .

Trilinear coordinates of Hofstadter -point

The trilinear coordinates of the Hofstadter -point are given below.

$$\backslash frac\; \backslash \; :\backslash \; \backslash frac\; \backslash \; :\backslash \; \backslash frac$$

Hofstadter zero- and one-points

The trilinear coordinates of these points cannot be obtained by plugging in the values 0 and 1 for in the expressions for the trilinear coordinates for the Hofstadter -point.

The Hofstadter zero-point is the limit of the Hofstadter -point as approaches zero; thus, the trilinear coordinates of Hofstadter zero-point are derived as follows:

Since

\lim _{r} \tfrac{\sin rA}{rA} = \lim _{r} \tfrac{\sin rB}{rB} = \lim _{r} \tfrac{\sin rC}{rC} = 1 ,

$$\backslash implies\; \backslash frac\backslash \; :\backslash \; \backslash frac\backslash \; :\backslash \; \backslash frac\; \backslash quad\; =\; \backslash quad\; \backslash frac\backslash \; :\backslash \; \backslash frac\backslash \; :\backslash \; \backslash frac$$

The Hofstadter one-point is the limit of the Hofstadter -point as approaches one; thus, the trilinear coordinates of the Hofstadter one-point are derived as follows:

Since

\lim _{r} \tfrac{( 1- r)A}{\sin(A - rA)} = \lim _{r} \tfrac{( 1- r)B}{\sin(B - rB)} = \lim _{r} \tfrac{( 1- r)C}{\sin(C - rC)} = 1 ,

$$\backslash implies\; \backslash frac\backslash \; :\backslash \; \backslash frac\backslash \; :\backslash \; \backslash frac\; \backslash quad\; =\; \backslash quad\; \backslash frac\backslash \; :\backslash \; \backslash frac\backslash \; :\backslash \; \backslash frac$$

Notes and References

1. Web site: Kimberling. Clark. Hofstadter points. 11 May 2012.
2. Web site: Weisstein. Eric W.. Hofstadter Triangle. MathWorld--A Wolfram Web Resource. 11 May 2012.
3. C. Kimberling. Hofstadter points. Nieuw Archief voor Wiskunde. 1994. 12. 109–114.