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Euler's quadrilateral theorem or Euler's law on quadrilaterals, named after Leonhard Euler (1707–1783), describes a relation between the sides of a convex quadrilateral and its diagonals. It is a generalisation of the parallelogram law which in turn can be seen as generalisation of the Pythagorean theorem .
A line will connect any two points, so a first degree polynomial equation is an exact fit through any two points with distinct x coordinates. If the order of the equation is increased to a second degree polynomial, the following results: = + +. This will exactly fit a simple curve to three points.
A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as d {\displaystyle d} approaches zero, a cyclic quadrilateral converges into a triangle A ′ B ′ C ′ , {\displaystyle \triangle A'B'C',} and the formulas above simplify to the analogous triangle formulas.
A polynomial function is one that has the form = + + + + + where n is a non-negative integer that defines the degree of the polynomial. A polynomial with a degree of 0 is simply a constant function; with a degree of 1 is a line; with a degree of 2 is a quadratic; with a degree of 3 is a cubic, and so on.
More generally, if the quadrilateral is a rectangle with sides a and b and diagonal d then Ptolemy's theorem reduces to the Pythagorean theorem. In this case the center of the circle coincides with the point of intersection of the diagonals. The product of the diagonals is then d 2, the right hand side of Ptolemy's relation is the sum a 2 + b 2.
Tangential quadrilateral with inradius r. The four line segments between the center of the incircle and the points where it is tangent to the quadrilateral partition the quadrilateral into four right kites. If a line cuts a tangential quadrilateral into two polygons with equal areas and equal perimeters, then that line passes through the ...
The two bimedians in a quadrilateral and the line segment joining the midpoints of the diagonals in that quadrilateral are concurrent and are all bisected by their point of intersection. [24]: p.125 In a convex quadrilateral with sides a, b, c and d, the length of the bimedian that connects the midpoints of the sides a and c is
A quadrilateral is a kite if and only if any one of the following conditions is true: The four sides can be split into two pairs of adjacent equal-length sides. [7] One diagonal crosses the midpoint of the other diagonal at a right angle, forming its perpendicular bisector. [9] (In the concave case, the line through one of the diagonals bisects ...