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Isogonal conjugate transformation over the points inside the triangle. In geometry, the isogonal conjugate of a point P with respect to a triangle ABC is constructed by reflecting the lines PA, PB, PC about the angle bisectors of A, B, C respectively. These three reflected lines concur at the isogonal conjugate of P.
The hyperbola is the isogonal conjugate of , the line joining the circumcenter and the incenter. [3] This fact leads to a few interesting properties. Specifically all the points lying on the line O I {\displaystyle OI} have their isogonal conjugates lying on the hyperbola.
In geometry, the isotomic conjugate of a point P with respect to a triangle ABC is another point, defined in a specific way from P and ABC: If the base points of the lines PA, PB, PC on the sides opposite A, B, C are reflected about the midpoints of their respective sides, the resulting lines intersect at the isotomic conjugate of P.
The Nagel point is the isotomic conjugate of the Gergonne point.The Nagel point, the centroid, and the incenter are collinear on a line called the Nagel line.The incenter is the Nagel point of the medial triangle; [2] [3] equivalently, the Nagel point is the incenter of the anticomplementary triangle.
Here is a definition of triangle geometry from 1887: "Being given a point M in the plane of the triangle, we can always find, in an infinity of manners, a second point M' that corresponds to the first one according to an imagined geometrical law; these two points have between them geometrical relations whose simplicity depends on the more or ...
The isogonal conjugate of each point X on the circumconic, other than A, B, C, is a point on the line u x + v y + w z = 0. {\displaystyle ux+vy+wz=0.} This line meets the circumcircle of ABC in 0,1, or 2 points according as the circumconic is an ellipse, parabola, or hyperbola.
The isogonal conjugate of the centroid X 2 is the symmedian point X 6 (also denoted by K) having trilinear coordinates a : b : c. So the Lemoine axis of ABC is the trilinear polar of the symmedian point of ABC. The tangential triangle of ABC is the triangle T A T B T C formed by the tangents to the circumcircle of ABC at its vertices.
If does not lie on the circumcircle then its isogonal conjugate yields the same pedal circle, that is the six points ,, and ,, lie on the same circle. Moreover, the midpoint of the line segment P Q {\displaystyle PQ} is the center of that pedal circle.