geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, 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 triangle's plane that is in some sense in the middle of the triangle. For example, the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, ...

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

Isoscelizer

An isoscelizer of an angle in a triangle is a line through points , where lies on and on , such that the triangle is an

isosceles triangle In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...

. An isoscelizer of angle is a line

perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...

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

Yff central triangle

Let be any triangle. Let be an isoscelizer of angle , be an isoscelizer of angle , and be an isoscelizer of angle . Let be the triangle formed by the three isoscelizers. The four triangles and are always similar. There is a unique set of three isoscelizers such that the four triangles and 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 modu ...

. In this special case formed by the three isoscelizers is called the Yff central triangle of . The

circumcircle In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumrad ...

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

Yff center of congruence

Let be any triangle. Let be the isoscelizers of the angles such that the triangle formed by them is the Yff central triangle of . The three isoscelizers are continuously parallel-shifted such that the three triangles are always congruent to each other until formed by the intersections of the isoscelizers reduces to a point. The point to which reduces to is called the Yff center of congruence of .

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 ...

of the Yff center of congruence are

\sec\frac : \sec\frac : \sec\frac

*Any triangle is the triangle formed by the lines which are externally tangent to the three excircles of the Yff central triangle of . *Let 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 bis ...

of . Let be the point on side such that , a point on side such that , and a point on side such that . Then the lines 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 in the plane of a triangle . Then points are taken on the sides such that

\angle BPD = \angle DPC, \quad \angle CPE = \angle EPA, \quad \angle APF = \angle FPB.

The generalization asserts that the lines are concurrent.

References