Triangles

Equilateral, Isosceles and Scalene. There are three special names given to triangles that tell how many sides (or angles) are equal. There can be 3, 2 or no.
Table of contents

Specifying an angle , a side , and an angle uniquely specifies a triangle with area.


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Given a triangle with two sides, the smaller and the larger, and one known angle , acute and opposite , if , there are two possible triangles. If , there is one possible triangle. If , there are no possible triangles. This is the ASS theorem.

Let be the base length and be the height. Finally, if all three sides are specified, a unique triangle is determined with area given by Heron's formula or by. This is the SSS theorem. In triangle geometry , it is frequently very convenient to use a triple of coordinates defined relative to the distances from each side of a given so-called reference triangle.

One form of such coordinates is known as trilinear coordinates , with all coordinates having the same sign corresponding to the triangle interior , one coordinate zero corresponding to a point on a side, two coordinates zero corresponding to a vertex, and coordinates having different signs corresponding to the triangle exterior. 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.

Triangle Classification

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.

Generalizing the partition of an acute triangle, any cyclic polygon that contains the center of its circumscribed circle can be partitioned into isosceles triangles by the radii of this circle through its vertices. The fact that all radii of a circle have equal length implies that all of these triangles are isosceles. This partition can be used to derive a formula for the area of the polygon as a function of its side lengths, even for cyclic polygons that do not contain their circumcenters.

This formula generalizes Heron's formula for triangles and Brahmagupta's formula for cyclic quadrilaterals. Either diagonal of a rhombus divides it into two congruent isosceles triangles. Similarly, one of the two diagonals of a kite divides it into two isosceles triangles, which are not congruent except when the kite is a rhombus. Isosceles triangles commonly appear in architecture as the shapes of gables and pediments. In Ancient Greek architecture and its later imitations, the obtuse isosceles triangle was used; in Gothic architecture this was replaced by the acute isosceles triangle.

Warren truss structures, such as bridges, are commonly arranged in isosceles triangles, although sometimes vertical beams are also included for additional strength. In graphic design and the decorative arts , isosceles triangles have been a frequent design element in cultures around the world from at least the Early Neolithic [42] [ original research? If a cubic equation with real coefficients has three roots that are not all real numbers , then when these roots are plotted in the complex plane as an Argand diagram they form vertices of an isosceles triangle whose axis of symmetry coincides with the horizontal real axis.

This is because the complex roots are complex conjugates and hence are symmetric about the real axis. In celestial mechanics , the three-body problem has been studied in the special case that the three bodies form an isosceles triangle, because assuming that the bodies are arranged in this way reduces the number of degrees of freedom of the system without reducing it to the solved Lagrangian point case when the bodies form an equilateral triangle.


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    • The first solutions of the problem with unbounded oscillations were found in the isosceles three-body problem. Long before isosceles triangles were studied by the ancient Greek mathematicians , the practitioners of Ancient Egyptian mathematics and Babylonian mathematics knew how to calculate their area. The theorem that the base angles of an isosceles triangle are equal appears as Proposition I. 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.

    References

    Robin Wilson credits this argument to Lewis Carroll , [51] who published it in , but W. Rouse Ball published it in and later wrote that Carroll obtained the argument from him.

    Triangle Song

    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.

    Area of triangles (article) | Khan Academy

    The triakis triangular tiling. Catalan solids with isosceles triangle faces. Obtuse isosceles pediment of the Pantheon, Rome. Flag of Saint Lucia.

    How to Calculate the Sides and Angles of Triangles

    See also Hadamard , Exercise , p. This also follows from the Exterior Angle Theorem. If one of the angles in a triangle is obtuse, the triangle is called obtuse. A triangle with one right angle is right. Otherwise, a triangle is acute ; for all of its angles are acute. All the definitions are naturally exclusive. There is no possible ambiguity. I came across this diagram in [ Jacobs , p. Contact Front page Contents Geometry.

    BREAKING DOWN 'Triangle'

    Triangle Classification The basic elements of any triangle are its sides and angles. As regard their sides, triangles may be Scalene all sides are different Isosceles two sides are equal Equilateral all three sides are equal And as regard their angles, triangles may be Acute all angles are acute Right one angle is right Obtuse one angle is obtuse Equiangular all angles are equal This applet requires Sun's Java VM 2 which your browser may perceive as a popup. The first two are of Greek and related origins; the word "equilateral" is of Latin origin: This is how the two approaches are distinguished with Venn diagrams: The following diagram summarizes all possible triangle configurations.