User:Clark Kimberling

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In triangle geometry, a circumconic is a conic section that passes through three given points, and an inconic is a conic section inscribed in the triangle of three given points.

Suppose A,B,C are distinct point, and let ΔABC denote the triangle whose vertices are A,B,C. Following common practice, A denotes not only the vertex but also the angle BAC at vertex A, and similarly for B and C as angles in ΔABC. Let a = |BC|, b = |CA|, c = |AB|, the sidelengths of ΔABC.

In trilinear coordinates, the general circumconic is the locus of a variable point X = x : y : z satisfying an equation

uyz + vzx + wxy = 0,

for some point u : v : w. The isogonal conjugate of each point X on the circumconic, other than A,B,C, is a point on the line

ux + vy + wz = 0.

This line meets the circumcircle of ΔABC in 0,1, or 2 points according as the circumconic is an ellipse, parabola, or hyperbola.

The general inconic is tangent to the three sidelines of ΔABC and is given by the equation

u²x² + v²y² + w²z² - 2vwyz - 2wuzx - 2uvxy = 0.

Contents 1 Centers and tangent lines 2 Other features 3 Examples 4 Lac qui Parle 5 Beatty's Theorem 6 Example 7 History 8 References 9 External Links

[edit] Centers and tangent lines

The center of the general circumconic is the point

u(-au + bv + cw) : v(au - bv + cw) : w(au + bv - cw).

The lines tangent to the general circumconic at the vertices A,B,C are, respectively,

wv + vz = 0,

uz + wx = 0,

vx + uy = 0.

The center of the general inconic is the point

cy + bz : az+ cx : bx + ay..

The lines tangent to the general inconic are the sidelines of ΔABC, given by the equations x = 0, y = 0, z = 0.

[edit] Other features

• Each noncircular circumconic meets the circumcircle of ΔABC in a point other than A, B, and C, often called the fourth point of intersection, given by trilinear coordinates

(cx - az)(ay - bx) : (ay - bx)(bz - cy) : (bz - cy)(cx - az)

• If P = p : q : r is a point on the general circumconic, then the line tangent to the conic at P is given by

(yr + zq)x + (zp + xr)y + (xq + yp)z = 0.

• The general circumconic reduces to a parabola if and only if

u²a² + v²b² + w²c² - 2vwbc - 2wuca - 2uvab = 0,

and to a rectangular hyperbola if and only if

x cos A + y cos B + z cos C = 0.

• The general inconic reduces to a parabola if and only if

ubc + yca + zab = 0.

• Suppose that p₁ : q₁ : r₁ and p₂ : q₂ : r₂ are distinct points, and let

X = (p₁ + p₂t) : (q₁ + q₂t) : (r₁ + r₂t).

As the parameter t ranges through the real numbers, the locus of X is a line. Define

X² = (p₁ + p₂t)² : (q₁ + q₂t)² : (r₁ + r₂t)².

The locus of X² is the inconic, necessarily an ellipse, given by the equation

L⁴x² + M⁴y² + N⁴z² - 2M²N²yz - 2N²L²zx - 2L²M²xy = 0,

where

L = q₁r₂ - r₁q₂,

M = r₁p₂ - p₁r₂,

N = p₁q₂ - q₁p₂.

[edit] Examples

Joseph Renville

Joseph Renville (1779-1846) was an interpreter for the 1805 Pike expedition and the 1823 Long expedition. He became an important figure in dealings between white men and Dakota (Sioux) Indians in Minnesota, as typified by his cooperation with missionaries. His hymnal Dakota dowanpi kin, "composed by J. Renville and sons, and the missionaries of the A.B.C.F.M." was published in Boston in 1842. Its successor, Dakota Odowan, is still in use today.

The town of Renville, Minnesota, is named in his honor, as are Renville County, Minnesota and Renville County, North Dakota.

Joseph Renville's father, also named Joseph Renvill, was a French Canadian fur trader, and his mother, Miniyuhe was a Dakota, possibly a daughter of Mdewakaton-0Dakota chief Big Thunder. Renville's bicultural formative years probably included some education in Canada.

[edit] Lac qui Parle

By 1827, Renville had settled at Lac qui Parle, Minnesota, where he built a stockade, kept a band of warriors, and continued his livelihood as a fur trader. In 1835, Thomas Smith Williamson, M.D. (1800-1879), arrived at Lac qui Parle. He was the first of several missionaries

Beatty Sequence

Suppose r and s are positive irrational numbers satisfying

1 / r + 1 / s = 1.

Then the sequences and are a pair of complementary Beatty sequences, and each is a Beatty sequence.

[edit] Beatty's Theorem

A pair of complementary Beatty sequences partition the set of positive integers.

Proof: We must show that every positive integer lies in one and only one of the two sequences and shall do so by considering the positions occupied by all the fractions j / r and r / s when they are jointly listed in nondecreasing order.

For any j / r, there are j numbers and numbers , so that the position of j / r in the list is . The equation 1 / r + 1 / s = 1 implies

.

Likewise, the position of k / s in the list is .

To see that no is also a , suppose to the contrary that j/r = k/s for some j and k. Then r/s = j/k, a rational number, but also, r/s = r(1 - 1/r) = r - 1, not a rational number.

Conclusion: every positive integer (that is, every position in the list) is in one and only one of the two Beatty sequences.

[edit] Example

For r = the golden ratio, we have s = r + 1. In this case, the , known as the lower Wythoff sequence, is

• (1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 25, 27, 29, ... )

and the sequence , the upper Wythoff sequence, is

• (2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, 36, 39, 41, 44, 47, ... ).

[edit] History

Beatty sequences solve a problem posed in the American Mathematical Monthly by Samuel Beatty in 1926. It has been cited many times — perhaps more than any other problem ever posed in the Monthly Problems section.

[edit] References

• Samuel Beatty, Problem 3172, American Mathematical Monthly 33 (1926) 159 and 34 (1927) 159.
• Kenneth Stolarsky, "Beatty sequences, continued fractions, and certain shift operators," Canadian Mathematical Bulletin 19 (1976) 473-482. Contains many references to Beatty sequences.

[edit] External Links

• Lower Wythoff sequence in The On-Line Encyclopedia of Integer Sequences
• Upper Wythoff sequence in The On-Line Encyclopedia of Integer Sequences
• Beatty Sequence at MathWorld