In plane geometry, an extended side or sideline of a polygon is the line that contains one side of the polygon. The extension of a finite side into an infinite line arises in various contexts.
In an obtuse triangle, the altitudes from the acute angled vertices intersect the corresponding extended base sides but not the base sides themselves.
The excircles of a triangle, as well as the triangle's inconics that are not inellipses, are externally tangent to one side and to the other two extended sides.
Trilinear coordinates locate a point in the plane by its relative distances from the extended sides of a reference triangle. If the point is outside the triangle, the perpendicular from the point to the sideline may meet the sideline outside the triangle—that is, not on the actual side of the triangle.
In a triangle, three intersection points, each of an external angle bisector with the opposite extended side, are collinear.
In a triangle, three intersection points, two of them between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended, are collinear.
An ex-tangential quadrilateral is a quadrilateral for which there exists a circle that is tangent to all four extended sides. The excenter (center of the tangent circle) lies at the intersection of six angle bisectors. These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors (supplementary angle bisectors) at the other two vertex angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect.
Pascal's theorem states that if six arbitrary points are chosen on a conic section (i.e., ellipse, parabola or hyperbola) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet in three points which lie on a straight line, called the Pascal line of the hexagon.
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An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's angles must sum to 180° in Euclidean geometry, no Euclidean triangle can have more than one obtuse angle. Acute and obtuse triangles are the two different types of oblique triangles — triangles that are not right triangles because they do not have a 90° angle.
In Euclidean geometry, a circumconic is a conic section that passes through the three vertices of a triangle, and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle. Suppose A, B, C are distinct non-collinear points, and let △ABC denote the triangle whose vertices are A, B, C. Following common practice, A denotes not only the vertex but also the angle ∠BAC at vertex A, and similarly for B and C as angles in △ABC. Let the sidelengths of △ABC.
In geometry, the trilinear coordinates x : y : z of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio x : y is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices A and B respectively; the ratio y : z is the ratio of the perpendicular distances from the point to the sidelines opposite vertices B and C respectively; and likewise for z : x and vertices C and A.
An integrated map of the genome of the tubercle bacillus, Mycobacterium tuberculosis, was constructed by using a twin-pronged approach. Pulsed-field gel electrophoretic analysis enabled cleavage sites for Asn I and Dra I to be positioned on the 4.4-Mb circ ...
Despite its inherent mechanical fragility, silica is widely used as a skeletal material in a great diversity of organisms ranging from diatoms and radiolaria to sponges and higher plants. In addition to their micro- and nanoscale structural regularity, man ...
Crank arrangement (1) for driving a pin-driven two degrees? of freedom mechanical oscillator (5), comprising:- a crank element (7) arranged to be rotationally driven about an axis of rotation (19) by means of a mechanical source of energy (M),- a crank slo ...