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The Kobon triangle problem is an unsolved problem in combinatorial geometry first stated by Kobon Fujimura (1903-1983). The problem asks for the largest number N(k) of nonoverlapping triangles whose sides lie on an arrangement of k lines.
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number , other examples being square numbers and cube numbers . The n th triangular number is the number of dots in the triangular arrangement with n dots on each side, and is equal to the sum of the n natural ...
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 =, where Roberts's theorem guarantees that three triangles will exist, but the solution to the Kobon triangle problem has five triangles.
After relating area to the number of triangles in this way, the proof concludes by using Euler's polyhedral formula to relate the number of triangles to the number of grid points in the polygon. [5] Tiling of the plane by copies of a triangle with three integer vertices and no other integer points, as used in the proof of Pick's theorem
Digon – 2 sides; Triangle – 3 sides Acute triangle; Equilateral triangle; Heptagonal triangle; Isosceles triangle. Golden Triangle; Obtuse triangle; Rational triangle; Heronian triangle. Pythagorean triangle; Isosceles heronian triangle; Primitive Heronian triangle; Right triangle. 30-60-90 triangle; Isosceles right triangle; Kepler ...
Since no triangle can have two obtuse angles, γ is an acute angle and the solution γ = arcsin D is unique. If b < c, the angle γ may be acute: γ = arcsin D or obtuse: γ ′ = 180° − γ. The figure on right shows the point C, the side b and the angle γ as the first solution, and the point C ′, side b ′ and the angle γ ′ as the ...
Triangulations have a number of applications, and there is an interest to find the "good" triangulations of a given point set under some criteria as, for instance minimum-weight triangulations. Sometimes it is desirable to have a triangulation with special properties, e.g., in which all triangles have large angles (long and narrow ("splinter ...
Te 12 = 364 is the total number of gifts "my true love sent to me" during the course of all 12 verses of the carol, "The Twelve Days of Christmas". [3] The cumulative total number of gifts after each verse is also Te n for verse n. The number of possible KeyForge three-house combinations is also a tetrahedral number, Te n−2 where n is the ...