geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the Yff center of congruence is a special point associated with a triangle. This special point is a

triangle center In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For example ...

and Peter Yff initiated the study of this triangle center in 1987. Yff center of congruence

Isoscelizer

An isoscelizer of an angle ''A'' in a triangle ''ABC'' is a line through points ''P''₁ and ''Q''₁, where ''P''₁ lies on ''AB'' and ''Q''₁ on ''AC'', such that the triangle ''AP''₁''Q''₁ is an

isosceles triangle In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...

. An isoscelizer of angle ''A'' is a line

perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...

to the bisector of angle ''A''. Isoscelizers were invented by Peter Yff in 1963.

Yff central triangle

Let ''ABC'' be any triangle. Let ''P''₁''Q''₁ be an isoscelizer of angle ''A'', ''P''₂''Q''₂ be an isoscelizer of angle ''B'', and ''P''₃''Q''₃ be an isoscelizer of angle ''C''. Let ''A'B'C' '' be the triangle formed by the three isoscelizers. The four triangles ''A'P₂Q₃'', ''Q₁B'P₃'', ''P₁Q₂C, and ''A'B'C' '' are always similar. There is a unique set of three isoscelizers ''P''₁''Q''₁, ''P''₂''Q''₂, ''P''₃''Q''₃ such that the four triangles ''A'''''P''₂''Q''₃, ''Q''₁''B'''''P''₃, ''P''₁''Q''₂''C''', and ''A'B'C' '' are

congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...

. In this special case the triangle ''A'B'C' '' formed by the three isoscelizers is called the Yff central triangle of triangle ''ABC''. The

circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

of the Yff central triangle is called the Yff central circle of the triangle.

Yff center of congruence

Let ''ABC'' be any triangle. Let ''P''₁''Q''₁, ''P''₂''Q''₂, ''P''₃''Q''₃ be the isoscelizers of the angles ''A'', ''B'', ''C'' such that the triangle ''A'B'C' '' formed by them is the Yff central triangle of triangle ''ABC''. The three isoscelizers ''P''₁''Q''₁, ''P''₂''Q''₂, ''P''₃''Q''₃ are continuously parallel-shifted such that the three triangles ''A'P₂Q₃'', ''Q₁B'P₃'', ''P₁Q₂C are always congruent to each other until the triangle ''A'B'C' '' formed by the intersections of the isoscelizers reduces to a point. The point to which the triangle ''A'B'C' '' reduces to is called the Yff center of congruence of triangle ''ABC''.

Properties

*The

trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...

of the Yff center of congruence are ( sec( ''A''/2 ) : sec ( ''B''/2 ), sec ( ''C''/2 ). *Any triangle ''ABC'' is the triangle formed by the lines which are externally tangent to the three excircles of the Yff central triangle of triangle ''ABC''. *Let ''I'' be the

incenter In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisec ...

of triangle ''ABC''. Let ''D'' be the point on side ''BC'' such that ∠''BID'' = ∠''DIC'', ''E'' a point on side ''CA'' such that ∠''CIE'' = ∠''EIA'', and ''F'' a point on side ''AB'' such that ∠''AIF'' = ∠''FIB''. Then the lines ''AD''. ''BE'', and ''CF'' are concurrent at the Yff center of congruence. This fact gives a geometrical construction for locating the Yff center of congruence. *A computer assisted search of the properties of the Yff central triangle has generated several interesting results relating to properties of the Yff central triangle. Yff Centre Generalisation

Generalization

The geometrical construction for locating the Yff center of congruence has an interesting generalization. The generalisation begins with an arbitrary point ''P'' in the plane of a triangle ''ABC''. Then points ''D'', ''E'', ''F'' are taken on the sides ''BC'', ''CA'', ''AB'' such that ∠''BPD'' = ∠''DPC'', ∠''CPE'' = ∠''EPA'', and ∠''APF'' = ∠''FPB''. The generalization asserts that the lines ''AD'', ''BE'', ''CF'' are concurrent.

References