Английская Википедия:Droz-Farny line theorem

Содержание

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Файл:Droz-Farny line.svg

The line through <math>A_0,B_0,C_0</math> is Droz-Farny line

In Euclidean geometry, the Droz-Farny line theorem is a property of two perpendicular lines through the orthocenter of an arbitrary triangle.

Let <math>T</math> be a triangle with vertices <math>A</math>, <math>B</math>, and <math>C</math>, and let <math>H</math> be its orthocenter (the common point of its three altitude lines. Let <math>L_1</math> and <math>L_2</math> be any two mutually perpendicular lines through <math>H</math>. Let <math>A_1</math>, <math>B_1</math>, and <math>C_1</math> be the points where <math>L_1</math> intersects the side lines <math>BC</math>, <math>CA</math>, and <math>AB</math>, respectively. Similarly, let Let <math>A_2</math>, <math>B_2</math>, and <math>C_2</math> be the points where <math>L_2</math> intersects those side lines. The Droz-Farny line theorem says that the midpoints of the three segments <math>A_1A_2</math>, <math>B_1B_2</math>, and <math>C_1C_2</math> are collinear.

The theorem was stated by Arnold Droz-Farny in 1899, but it is not clear whether he had a proof.

Goormaghtigh's generalization

A generalization of the Droz-Farny line theorem was proved in 1930 by René Goormaghtigh.

As above, let <math>T</math> be a triangle with vertices <math>A</math>, <math>B</math>, and <math>C</math>. Let <math>P</math> be any point distinct from <math>A</math>, <math>B</math>, and <math>C</math>, and <math>L</math> be any line through <math>P</math>. Let <math>A_1</math>, <math>B_1</math>, and <math>C_1</math> be points on the side lines <math>BC</math>, <math>CA</math>, and <math>AB</math>, respectively, such that the lines <math>PA_1</math>, <math>PB_1</math>, and <math>PC_1</math> are the images of the lines <math>PA</math>, <math>PB</math>, and <math>PC</math>, respectively, by reflection against the line <math>L</math>. Goormaghtigh's theorem then says that the points <math>A_1</math>, <math>B_1</math>, and <math>C_1</math> are collinear.

The Droz-Farny line theorem is a special case of this result, when <math>P</math> is the orthocenter of triangle <math>T</math>.

Dao's generalization

The theorem was further generalized by Dao Thanh Oai. The generalization as follows:

First generalization: Let ABC be a triangle, P be a point on the plane, let three parallel segments AA', BB', CC' such that its midpoints and P are collinear. Then PA', PB', PC' meet BC, CA, AB respectively at three collinear points.

Файл:Daotheoremonconic1.svg

Dao's second generalization

Second generalization: Let a conic S and a point P on the plane. Construct three lines d_a, d_b, d_c through P such that they meet the conic at A, A'; B, B'  ; C, C' respectively. Let D be a point on the polar of point P with respect to (S) or D lies on the conic (S). Let DA' ∩ BC =A₀; DB' ∩ AC = B₀; DC' ∩ AB= C₀. Then A₀, B₀, C₀ are collinear.