In geometry, the Yff center of congruence is a special point associated with a triangle. This special point is a triangle center and initiated the study of this triangle center in 1987.[1]

Yff central triangle of triangle ABC

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. An isoscelizer of angle *A* is a line perpendicular to the bisector of angle *A*. Isoscelizers were invented by Peter Yff in 1963.^{[2]}

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'P _{2}Q_{3}*,

*Q*,

_{1}B'P_{3}*P*, and

_{1}Q_{2}C'*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. In this special case the triangle *A'B'C' *formed by the three isoscelizers is called the **Yff central triangle** of triangle *ABC*.^{[3]}

The circumcircle of the Yff central triangle is called the **Yff central circle** of the triangle.

Yff center of congruence

Animation showing the continuous shrinking of the Yff central triangle to the Yff center of congruence. The animation also shows the continuous expansion of the Yff central triangle until the three outer triangles reduce to points on the sides of the triangle.

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'P _{2}Q_{3}*,

*Q*,

_{1}B'P_{3}*P*are always congruent to each other until the triangle

_{1}Q_{2}C'*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

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.

- The trilinear coordinates of the Yff center of congruence are ( sec(
*A*/2 ) : sec (*B*/2 ), sec (*C*/2 ).^{[1]} - 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 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.^{[4]} - 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.
^{[5]}

Generalization of Yff centre of congruence

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.^{[4]}
See also

Congruent isoscelizers point

References

Kimberling, Clark. "Yff Center of Congruence". Retrieved 30 May 2012.

Weisstein, Eric W. "Isoscelizer". MathWorld--A Wolfram Web Resource. Retrieved 30 May 2012.

Weisstein, Eric W. "Yff central triangle". MathWorld--A Wolfram Web Resource. Retrieved 30 May 2012.

Kimberling, Clark. "X(174) = Yff Center of Congruence". Retrieved 2 June 2012.

Dekov, Deko (2007). "Yff Center of Congruence". Journal of Computer-Generated Euclidean Geometry. 37: 1–5. Retrieved 30 May 2012.

Undergraduate Texts in Mathematics

Graduate Studies in Mathematics

Hellenica World - Scientific Library

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