The first two are of Greek and related origins; the word "equilateral" is of Latin origin:. A scalene triangle is uneven in the sense that all three sides are of different lengths. The scalene muscles on each side of a person's neck are named for their triangular appearance.
A scalene cone or cylinder is one whose axis is not perpendicular to its base; opposite elements make "uneven" angles with the base. The Indo-European root s kel- "curved, bent" is found in scoliosis and colon , borrowed from Greek. In geometry, an isosceles triangle or trapezoid has two equal legs.
It may seem strange that the root means "bent" even though the sides of a triangle or trapezoid are straight, but each leg is bent relative to the adjoining legs. Related borrowings from Latin are bilateral and multilateral. In geometry, equilateral triangle is one in which all sides are equal in length. This is how the two approaches are distinguished with Venn diagrams:.
As regard the angles, a triangle is equiangular if all three of its angles are equal. Very early in the Elements I. From here, for a triangle, the properties of being equilateral and equiangular are equivalent, and the latter is seldom mentioned. Rival explanations for this name include the theory that it is because the diagram used by Euclid in his demonstration of the result resembles a bridge, or because this is the first difficult result in Euclid, and acts to separate those who can understand Euclid's geometry from those who cannot.
A well known fallacy is the false proof of the statement that all triangles are isosceles. Robin Wilson credits this argument to Lewis Carroll ,  who published it in , but W. Rouse Ball published it in and later wrote that Carroll obtained the argument from him. From Wikipedia, the free encyclopedia. For other uses, see Isosceles disambiguation.
Three congruent inscribed squares in the Calabi triangle. A golden triangle subdivided into a smaller golden triangle and golden gnomon. The triakis triangular tiling. Catalan solids with isosceles triangle faces. Obtuse isosceles pediment of the Pantheon, Rome. Flag of Saint Lucia. See also Hadamard , Exercise , p.
Although "many of the early Egyptologists" believed that the Egyptians used an inexact formula for the area, half the product of the base and side, Vasily Vasilievich Struve championed the view that they used the correct formula, half the product of the base and height Clagett Alsina, Claudi; Nelsen, Roger B.
Rouse ; Coxeter, H. Isoperimetric problems and the origin of the quadratic equations", Isis , See in particular p. Visual Mathematics , Boston: June , " July , "A study of the red on cream and cream on red designs on Early Neolithic ceramics from Nea Nikomedeia", American Journal of Archaeology , 88 3: Wilson, Robin , Lewis Carroll in Numberland: His fantastical mathematical logical life, an agony in eight fits , Penguin Books, pp.
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In other projects Wikimedia Commons. This page was last edited on 14 September , at The straightedge and compass construction of the triangle can be accomplished as follows. In the above figure, take as a radius and draw.
Then bisect and construct. Extending to locate then gives the equilateral triangle. Another construction proceeds by drawing a circle of the desired radius centered at a point. Choose a point on the circle's circumference and draw another circle of radius centered at. The two circles intersect at two points, and , and is the second point at which the line intersects the first circle.
Unlike a general polygon with sides, a triangle always has both a circumcircle and an incircle. A triangle with sides , , and can be constructed by selecting vertices 0, 0 , , and , then solving. The angles of a triangle satisfy the law of cosines.
The latter gives the pretty identity. Trigonometric functions of half angles in a triangle can be expressed in terms of the triangle sides as. Let stand for a triangle side and for an angle, and let a set of s and s be concatenated such that adjacent letters correspond to adjacent sides and angles in a triangle.
In each of these cases, the unknown three quantities there are three sides and three angles total can be uniquely determined.
Other combinations of sides and angles do not uniquely determine a triangle: Dividing the sides of a triangle in a constant ratio and then drawing lines parallel to the adjacent sides passing through each of these points gives line segments which intersect each other and one of the medians in three places. If , then the extensions of the side parallels intersect the extensions of the medians. The medians bisect the area of a triangle, as do the side parallels with ratio.
The envelope of the lines which bisect the area a triangle forms three hyperbolic arcs. The envelope is somewhat more complicated, however, for lines dividing the area of a triangle into a constant but unequal ratio Dunn and Petty , Ball , Wells There are four circles which are tangent to the sides of a triangle, one internal the incircle and the rest external the excircles. Their centers are the points of intersection of the angle bisectors of the triangle. Any triangle can be positioned such that its shadow under an orthogonal projection is equilateral. Introduction to Geometry, 2nd ed.
Monthly , , The Straight Line and Circle. Riegel und Wiesner,