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From any point on the circumcircle of a triangle, the nearest points on each of the three extended sides of the triangle are collinear in the Simson line of the point on the circumcircle. The lines connecting the feet of the altitudes intersect the opposite sides at collinear points. [3]: p.199
The concept was first published, however, by William Wallace in 1799, [3] and is sometimes called the Wallace line. [ 4 ] The converse is also true; if the three closest points to P on three lines are collinear, and no two of the lines are parallel, then P lies on the circumcircle of the triangle formed by the three lines.
The three types of degenerate triangles, all of which contain zero area. A degenerate triangle is a "flat" triangle in the sense that it is contained in a line segment. It has thus collinear vertices [3] and zero area. If the three vertices are pairwise distinct, it has two 0° angles and one 180° angle.
Three or more collinear points, where the circumcircles are of infinite radii. Four or more points on a perfect circle, where the triangulation is ambiguous and all circumcenters are trivially identical. In this case the Voronoi diagram contains vertices of degree four or greater and its dual graph contains polygonal faces with four or more sides.
The left-hand side of this equation is a vector that has the same direction as the line CF, and the right-hand side has the same direction as the line AB. These lines have different directions since A, B, C are not collinear. It follows that the two members of the equation equal the zero vector, and
This maximum is attained for simple arrangements, those in which each two lines cross at a vertex that is disjoint from all the other lines. The number of vertices is smaller when some lines are parallel, or when some vertices are crossed by more than two lines. [4] An arrangement can be rotated, if necessary, to avoid axis-parallel lines.
The ten lines involved in Desargues's theorem (six sides of triangles, the three lines Aa, Bb and Cc, and the axis of perspectivity) and the ten points involved (the six vertices, the three points of intersection on the axis of perspectivity, and the center of perspectivity) are so arranged that each of the ten lines passes through three of the ...
This happens if and only if the triangle vertices aren't collinear and the ray isn't parallel to the plane. The algorithm can use Cramer's Rule to find the t {\displaystyle t} , u {\displaystyle u} , and v {\displaystyle v} values for an intersection, and if it lies within the triangle, the exact coordinates of the intersection can be found by ...