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The Schiffler point of a triangle is the point of concurrence of the Euler lines of four triangles: the triangle in question, and the three triangles that each share two vertices with it and have its incenter as the other vertex. The Napoleon points and generalizations of them are points of concurrency. For example, the first Napoleon point is ...
Ceva's theorem, case 1: the three lines are concurrent at a point O inside ABC Ceva's theorem, case 2: the three lines are concurrent at a point O outside ABC. In Euclidean geometry, Ceva's theorem is a theorem about triangles.
The theorem claims that the three straight lines, , and are concurrent. [1] This result was established by the Romanian mathematician Cezar Coşniţă (1910-1962). [2] Their point of concurrence is known as the triangle's Kosnita point (named by Rigby in
A triangle ABC with the incenter I has its Schiffler point at the point of concurrence of the Euler lines of the four triangles BCI, CAI, ABI, ABC. Schiffler's theorem states that these four lines all meet at a single point.
In geometry, the Tarry point T for a triangle ABC is a point of concurrency of the lines through the vertices of the triangle perpendicular to the corresponding sides of the triangle's first Brocard triangle DEF. The Tarry point lies on the other endpoint of the diameter of the circumcircle drawn through the Steiner point. [1]
The two triangles on the left are congruent. The third is similar to them. The last triangle is neither congruent nor similar to any of the others. Congruence permits alteration of some properties, such as location and orientation, but leaves others unchanged, like distances and angles. The unchanged properties are called invariants.
The incenter is the Nagel point of the medial triangle; [2] [3] equivalently, the Nagel point is the incenter of the anticomplementary triangle. The isogonal conjugate of the Nagel point is the point of concurrency of the lines joining the mixtilinear touchpoint and the opposite vertex.
In Euclidean geometry, the Apollonius point is a triangle center designated as X(181) in Clark Kimberling's Encyclopedia of Triangle Centers (ETC). It is defined as the point of concurrence of the three line segments joining each vertex of the triangle to the points of tangency formed by the opposing excircle and a larger circle that is tangent to all three excircles.