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In particular, for three points in the plane (n = 2), the above matrix is square and the points are collinear if and only if its determinant is zero; since that 3 × 3 determinant is plus or minus twice the area of a triangle with those three points as vertices, this is equivalent to the statement that the three points are collinear if and only ...
Infinitely many triangles have the same angles, since specifying the angles of a triangle does not determine its size. (A degenerate triangle, whose vertices are collinear, has internal angles of 0° and 180°; whether such a shape counts as a triangle is a matter of convention.
The Euler lines of the 10 triangles with vertices chosen from A, B, C, F 1 and F 2 are concurrent at the centroid of triangle ABC. [ 12 ] The Euler lines of the four triangles formed by an orthocentric system (a set of four points such that each is the orthocenter of the triangle with vertices at the other three points) are concurrent at the ...
Desargues's theorem states that the truth of the first condition is necessary and sufficient for the truth of the second. In projective geometry, Desargues's theorem, named after Girard Desargues, states: Two triangles are in perspective axially if and only if they are in perspective centrally. Denote the three vertices of one triangle by a, b ...
Then I and J are collinear with one of the three triangle vertices. The circle with IJ as diameter passes through the other two vertices and is centered on the circumcircle of ABC. When one of I or J is the incenter, this is the trillium theorem, with line IJ as the (internal) angle bisector of one of the triangle's angles.
Thales's theorem. Angle formed by a point on a circle and the 2 ends of a diameter is a right angle. Thales’ theorem: if AC is a diameter and B is a point on the diameter's circle, the angle ∠ ABC is a right angle. In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the ...
A vertex of an angle is the endpoint where two lines or rays come together. In geometry, a vertex (pl.: vertices or vertexes) is a point where two or more curves, lines, or edges meet or intersect. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedron are vertices. [1][2][3]
Simson line. In geometry, given a triangle ABC and a point P on its circumcircle, the three closest points to P on lines AB, AC, and BC are collinear. [1] The line through these points is the Simson line of P, named for Robert Simson. [2] The concept was first published, however, by William Wallace in 1799, [3] and is sometimes called the ...