X₁₂₈₁

3rd SHARYGIN POINT

The Sharygin points are described in

5, 6, 7 3, 4, 5 Kepler Kimberling

X₁₂₈₁ (3rd SHARYGIN POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {21, 99}, {98, 100}

• Related Centers: X₂₁ (SCHIFFLER POINT), X₉₉ (STEINER POINT), X₉₈ (TARRY POINT), X₁₀₀ (ANTICOMPLEMENT OF FEUERBACH POINT)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc(a^{2} - bc)(b^{3} + c^{3} - a^{3} - abc)\)

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(3.337525171124\)

X₁₂₈₁ lies on the Yff contact circle.

▸ Incident Central Lines8 lines

{2, 846}, {21, 99}, {63, 147}, {98, 100}, {256, 291}, {350, 1284}, {385, 740}, {659, 804}

▸ Functional Relations8 relations

• X₁₂₈₁ = points of concurrence of lines DD', EE', FF'

• X₈₄₆ = homothetic center of the excentral triangle and triangle DEF

• X₁₀₅₄ = center of similitude of the excentral triangle and triangle D'E'F'.

• X₁₂₈₁ = isogonal conjugate of X₃₀₆₄₈

• X₁₂₈₁ = perspector of unary cofactor triangles of Gemini triangles 33 and 34

• X₁₂₈₁ = crossdifference of every pair of points on line X₅₀₂₉X_30654

• X₁₂₈₁ = X₁-of-1st-anti-Brocard-triangle

• X₁₂₈₁ = perspector of (cross-triangle of ABC and 1st Sharygin triangle) and (cross-triangle of ABC and 2nd Sharygin triangle)

Notes

Darij Grinberg, Sharygin Points Report, Hyacinthos #6293 (1/8/03) and #6315 (1/10/03)

The first of ten sections is an Introduction quoted, in part, here:

We will treat two remarkable triangles: the triangle bounded by the perpendicular bisectors of the internal angle bisectors of a triangle ABC, and the triangle bounded by the perpendicular bisectors of the external angle bisectors of triangle ABC. These two triangles and the triangle ABC are three perspective triangles, having a common perspectrix: the Lemoine axis of ABC. The mutual perspectors of the three triangles will be called the first, second and third Sharygin points of ABC (after a problem of Igor Sharygin - see Section 10). The report introduces fifteen Sharygin points, of which the 1st, 2nd, 4th, and 6th are X₂₅₆, X₂₉₁, X₈₄₆,

• and X₁₀₅₄, respectively. X₁₂₈₁ is the 3rd Sharygin point. See also Hyacinthos #6293 and #6315.

Let A' be the point where the internal angle bisector of angle CAB meets line BC, and let A″ be the point where the external angle bisector of angle CAB meets line BC. Let x be the perpendicular bisector of segment AA', and let x’ be the perpendicular bisector of segment AA″. Define y, z, y’, z’ cyclically. Let D be the point where lines y and z meet, and let D' be the point where lines y’ and z’ meet. Define E, F, E', F' cyclically. Then