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Marden's theorem states their location within this triangle more precisely: Suppose the zeroes z 1 , z 2 , and z 3 of a third-degree polynomial p ( z ) are non-collinear. There is a unique ellipse inscribed in the triangle with vertices z 1 , z 2 , z 3 and tangent to the sides at their midpoints : the Steiner inellipse .
For Kuratowski's theorem, the notion of containment is that of graph homeomorphism, in which a subdivision of one graph appears as a subgraph of the other. Thus, every graph either has a planar drawing (in which case it belongs to the family of planar graphs) or it has a subdivision of at least one of these two graphs as a subgraph (in which ...
Visual proof of Lee Sallows's triangle theorem. In 2014 Sallows discovered a previously unnoticed result, a way of using the medians to divide any triangle into three smaller triangles, all congruent with one another. Repeating the process on each triangle yields triangles similar to the original but a ninth the area. [10]
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 ...
Ceva's theorem can be obtained from it by setting the area equal to zero and solving. The analogue of the theorem for general polygons in the plane has been known since the early nineteenth century. [9] The theorem has also been generalized to triangles on other surfaces of constant curvature. [10]
The area of a triangle can be demonstrated, for example by means of the congruence of triangles, as half of the area of a parallelogram that has the same base length and height. A graphic derivation of the formula T = h 2 b {\displaystyle T={\frac {h}{2}}b} that avoids the usual procedure of doubling the area of the triangle and then halving it.
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros", or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields.
Here is a definition of triangle geometry from 1887: "Being given a point M in the plane of the triangle, we can always find, in an infinity of manners, a second point M' that corresponds to the first one according to an imagined geometrical law; these two points have between them geometrical relations whose simplicity depends on the more or ...