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In logical argument and mathematical proof, the therefore sign, ∴, is generally used before a logical consequence, such as the conclusion of a syllogism. The symbol consists of three dots placed in an upright triangle and is read therefore. While it is not generally used in formal writing, it is used in mathematics and shorthand.
The converse of the theorem is true as well. That is if a line is drawn through the midpoint of triangle side parallel to another triangle side then the line will bisect the third side of the triangle. The triangle formed by the three parallel lines through the three midpoints of sides of a triangle is called its medial triangle.
Whereas Roberts's theorem concerns the fewest possible triangles made by a given number of lines, the related Kobon triangle problem concerns the largest number possible. [3] The two problems differ already for n = 5 {\displaystyle n=5} , where Roberts's theorem guarantees that three triangles will exist, but the solution to the Kobon triangle ...
For example, a triangle (the 2-dimensional simplex) one 2-dimensional element (itself), three 1-dimensional elements (lines, or edges), and three 0-dimensional elements (vertices, or corners); this corresponds to the third row 1, 3, 3, 1 of Pascal's triangle. This fact can be explained by combining Pascal's rule for generating the triangle with ...
This distribution for a = 0, b = 1 and c = 0.5—the mode (i.e., the peak) is exactly in the middle of the interval—corresponds to the distribution of the mean of two standard uniform variables, that is, the distribution of X = (X 1 + X 2) / 2, where X 1, X 2 are two independent random variables with standard uniform distribution in [0, 1]. [1]
(The triangle A'B'C' is the Brocard triangle of triangle ABC.) Let L A be the line through A parallel to the line B'C', L B be the line through B parallel to the line C'A' and L C be the line through C parallel to the line A'B'. Then the three lines L A, L B and L C are concurrent. The point of concurrency is the Steiner point of triangle ABC.
The triple bar or tribar, ≡, is a symbol with multiple, context-dependent meanings indicating equivalence of two different things. Its main uses are in mathematics and logic. It has the appearance of an equals sign = with a third line.
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 ...