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Topic: Equilateral triangles

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	Triangle - Wikipedia, the free encyclopedia
		A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments.
		A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas.
		The area S of a triangle is S = ½bh, where b is the length of any side of the triangle (the base) and h (the altitude) is the perpendicular distance between the base and the vertex not on the base.
	en.wikipedia.org /wiki/Triangle (2072 words)

	Spherical structural arrangement - Patent 4026078 (Site not responding. Last check: 2007-11-06)
		Identical equilateral triangles B are then formed, the bases 24 of the triangles A being adjacent and equal to one side of each of the triangles B. The other two sides of the equilateral triangles B, being of equal length, are also designated 24 and form vertices 27.
		The A triangles are isosceles and utilize the two equal sides 22, the first basic element length, and share with the equilateral triangles B the common side 24, the second basic element length.
		The C triangles are also isosceles and utilize the equal sides 24 of the adjacent equilateral triangles B. The third side 32 on the base of the triangle C is the third basic element length.
	www.freepatentsonline.com /4026078.html (2251 words)

	Geometry :: 6-9 :: Chapter Three (Site not responding. Last check: 2007-11-06)
		Thus the three outside triangles are congruent to the unit triangle, and the hexagon is equal to twice the equilateral formed by the three yellow triangles nearby.
		As usual, examine the relationship between the lines of the hexagon and the equilateral triangle; the lines of the hexagon and those of the rhombus; the lines of the rhombus and the equilateral triangle (the diagonals are the only lines considered).
		Observe that the equilateral triangle is inscribed in the hexagon.
	www.moteaco.com /albums/geometry3.html (4177 words)

	section5
		Finally, in a triangle ABC a segment that connects the vertex B and the middle point of the side opposite to the vertex B is called the median falling from the vertex B.
		Theorem 5.3 In an isosceles triangle a median falling from the vertex angle is also the bisector of the vertex angle and the altitude falling from the vertex angle.
		Triangles ACD and BCD are equal by SAS: AD = DB since D is the middle point of AB, AC = CB since the triangle ABC is an isosceles triangle and angles A and B are equal as base angles of an isosceles triangle ABC.
	www.math.psu.edu /geom/koltsova/section5.html (781 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		If an equilateral triangle is dissected into a finite number of smaller equilateral triangles, then two of the smaller triangles are congruent.
		If an equilateral triangle is dissected into equilateral triangles, then we can obtain a dissection of a parallelogram into equilateral triangles by attaching another (undivided) equilateral triangle along one of the sides of the original triangle.
		But the triangle that we attached is larger than all of the constituent triangles of the original dissection.
	www.math.niu.edu /~rusin/known-math/99/equilateral (419 words)

	Triangles
		Triangles can be classified according to the length of their sides and the size of their angles.
		All angles of an equilateral triangle are 60°.
		To show that the sides of an equilateral triangle have the same length, we place identical marks on the sides of the triangle.
	www.mathsteacher.com.au /year7/ch09_polygons/01_triangle/tri.htm (317 words)

	Napoleon's Theorem
		If we construct equilateral triangles on any two of the sides of the given triangle, the distance from their centers is the same, regardless of which two sides we choose.
		The center of the equilateral triangle constructed on the side a is to be found by moving from the midpoint of that side perpendicularly outward a distance of a/(12)
		Recall that the edge lengths of a triangle are proportional to the sines of the opposite angles, which implies that the lengths of the segments from the vertex to the centroid of the equilateral triangles centered on points a,b,c are proportional to sin(
	www.mathpages.com /home/kmath270/kmath270.htm (1847 words)

	Basics (Site not responding. Last check: 2007-11-06)
		Thus, the maximum number of equilateral triangles comprising a spherical surface is 20 which is related to the icosahedron (formed by connecting the vertices of each equilateral triangle with a straight line).
		An examination of the sides of the smaller triangles in the least common denominator (LCD) triangle indicates that there are 22 different triangles for the frequency of 10 and each triangle in the LCD is different.
		Thus, the number of different triangles is proportional to the square of the frequency and is approximately the frequency squared divided by six.
	members.cox.net /mdavister/Basics.htm (1708 words)

	Definition of a Triangle
		Obtuse triangles are triangles in which the measure of one angle is greater than 90 degrees.
		Equilateral triangles are triangles in which all three sides are the same length.
		Scalene triangles are triangles in which none of the sides are the same length.
	www.geom.uiuc.edu /~wanous/definitionofatriangle.html (212 words)

	Cool math .com - Types of Triangles
		In the equilateral triangle, all the sides are the same length (congruent) and all the angles are the same size (congruent).
		The isosceles triangle (I can NEVER remember how to spell isosceles) has two sides that are the same length (congruent) and two angles that are the same size (congruent).
		Acute triangles have NO angles greater than or equal to 90 degrees -- all their angles are less than 90 degrees.
	www.coolmath.com /reference/triangles-types.html (249 words)

	Does isosceles subsume equilateral? \| Ask MetaFilter
		Mathworld says that an equilateral triangle is a special case of an isosceles triangle.
		So, a triangle with 3 equal sides is called equilateral, triangles with 2 equal sides are isosceles and finally a triangle with no equal sides is called scalene.
		Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.
	ask.metafilter.com /mefi/15865 (1344 words)

	Triangle of Medians (Site not responding. Last check: 2007-11-06)
		The sides of the triangles of medians were also measured in GSP to verify that the triangles of medians are equilateral.
		Looking at these constructions the triangles of medians appear to be right triangles, but when the angles are measured it is discovered they are close but not actually right triangles.
		We notice when both the original triangle and the triangle of medians are right triangles the other two angles measure 35 degrees and 55 degrees.
	jwilson.coe.uga.edu /emt668/EMAT6680.2004.SU/Stein/assign6/assign6.html (340 words)

	JIAN Xin-hua (Site not responding. Last check: 2007-11-06)
		Mechanically drawing the lacking lines (which are wont to be provided by the perceiving eye) between the edges of the smaller square to complete the larger squares and between the smaller equilateral triangles to complete the larger equilateral triangles produces a network that accommodates a two-color map treatment.
		Since numerous squares and equilateral triangles are embedded in the reticulation of this compound spiral, only one of the three emergent squares and only one of the four emergent triangles are, through coloration, elaborated at each level.
		The triangles are similarly treated, excepting rotated, interval by interval, in a counterclockwise manner.
	www.mi.sanu.ac.yu /vismath/huff/huf3.htm (300 words)

	THE FLAT UNIVERSE result of the relation between Equilateral Triangles
		The unit is the difference between the diameter delimited by the vertexes of an Equilateral triangle and the height of the same triangle.
		We are going to take 1/3 of the height of an equilateral triangle as Universal measurement unit and relate it with the principal readings that Physicist handles and see the coincidences among them.
		Where 3 is the triangle height as well as the number of vertexes and 2 corresponds to the triangle internal diameter.
	www.flat-universe.com (669 words)

	SparkNotes: Special Triangles: Classifying Triangles
		In triangle ABC above, side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. Angle A is included in sides b and c, angle B is included in sides a and c, and angle C is included in sides a and b.
		When triangles are classified according to the lengths of their sides, they fit into one of three categories: scalene, isosceles, or equilateral.
		When triangles are classified by their angles, they fit into one of four categories: acute, obtuse, right, or equilateral.
	www.sparknotes.com /math/geometry2/specialtriangles/section1.html (370 words)

	July 2002 solution (Site not responding. Last check: 2007-11-06)
		T is not equilateral, while the largest equilateral triangle that rests on side a is the same size as the largest that rest on sides b and c.
		The introductory comments tell us (and it is easy to show) that the largest equilateral triangle that fits in T sits on either side a or side c; the only way an equilateral triangle on the middle side b could be largest would be when b = a or b = c.
		Since the given triangle T is not equilateral, the only possibility that remains is for the two largest sides to be equal; that is, a = b.
	mathcentral.uregina.ca /MP/previous2001/july02sol.html (583 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Definition: An equilateral polygon is one whose sides all have the same length.
		A regular polygon is a polygon that is both equilateral and equiangular.
		Construct a tessellation of the plane by equilateral triangles.
	www.math.csusb.edu /courses/m129/eqtri.html (513 words)

	Pop Quiz: Further Information for Question #5
		An equilateral triangle is simply a triangle in which all three sides are of equal lengths.
		Since there are 180 degrees in a triangle, and all three angles are the same in an equilateral triangle, each angle of an equilateral triangle measures 60 degrees.
		An equilateral triangle is known as a regular polygon, since all its sides are equal and all its angles are equal.
	www.hamilton.net /subscribers/brian/popquiz/0005.html (623 words)

	Constructive Triangles
		One yellow equilateral triangle divided into three equal pieces by lines drawn from each angle to the center of the triangle (along angle bisectors).
		There are fl guidelines along the perimeter of the equilateral triangle and the bases of the smaller triangles.
		The child takes the equilateral triangles and discovers that there is only one shape to be made, no matter how he joins them.
	www.montessoriworld.org /sensory/sconstr.html (778 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Right triangles will never have an angle greater than 90 degrees.¡ó¨Isosceles Triangles¨¾A isosceles triangle is a triangle that has two sides that are the same size.
		In addition to having three sides of different length, a scalene triangle has angles that are different in length as well.ó ¨Equilateral Triangles¨°Equilateral triangles are triangles in which all three sides are the same length.
		An equilateral triangle will never have a right angle.ó¨Acute Triangle¨¶An acute triangle is a triangle that has no angles greater than 89 degrees.
	www2.gvsu.edu /~holdenl/ppointproject.ppt (354 words)

	Practice with Similarity
		The sides of a triangle are 5, 6 and 10.
		Find the length of the longest side of a similar triangle whose shortest side is 15.
		Similar triangles are exactly the same shape and size.
	regentsprep.org /Regents/math/similar/PracSim.htm (199 words)

	Pythagoras of Samos - Activity 7
		We will start the investigation by laying three of the triangles on our table so that each triangle is touching at a common point and so that adjacent sides are also touching (as in Activity 5).
		The three triangles will close in on each other forming a closed three faced shape with an open triangular base - they form a solid which has as its exposed base another equilateral triangle.
		Now, since each interior angle of an equilateral triangle is 60°, we could fit together three, four, or five of them at a vertex, and these correspond to to the tetrahedron, the octahedron, and the icosahedron.
	www.mathgym.com.au /history/pythagoras/Activity7.htm (1582 words)

	Sierpinski Gasket
		Divide this into four smaller equilateral triangles using the midpoints of the three sides of the original triangle as the new vertices.
		The Sierpinski gasket, also known as the Sierpinski triangle, is the intersection of all the sets in this sequence, that is, the set of points that remain after this construction is repeated infinitely often.
		Because of the rotational symmetry of an equilateral triangle, there is second iterated function system that has the Sierpinski gasket as its invariant set.
	ecademy.agnesscott.edu /~lriddle/ifs/siertri/siertri.htm (818 words)

	No Equilateral Triangles on a Chess Board (Site not responding. Last check: 2007-11-06)
		To show that that answer is no, assume an equilateral triangle whose verticies all have integer coordinates.
		Then we have the equilateral condition a^2 + b^2 = c^2 + d^2 = (a-c)^2 + (b-d)^2 = Q where Q is the square of the edge length.
		Now we can assert that the equilateral triangle cannot have all its verticies as lattice points, because any triangle whose verticies have integer coordinates must have rational area, as is clear from the area formula (x2-x3)(y1-y2) - (x1-x2)(y2-y3) A = ------------------------------- 2 For more on this formula, see Net Area and Green's Theorem
	www.mathpages.com /home/kmath261.htm (181 words)

	Congruence
		If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.
		of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
		You can draw 2 equilateral triangles that are the same shape but not the same size.
	regentsprep.org /Regents/math/congruen/Ltriangles.htm (255 words)

	American Mathematical Monthly, The: How many unit equilateral triangles can be generated by N points in convex position? (Site not responding. Last check: 2007-11-06)
		An equilateral triangle of side length one is called a unit triangle.
		If xyz is a clockwise-oriented unit triangle spanned by S, then xy is said to be a left edge with respect to x and a right edge with respect to y.
		Let xyz be a clockwise-oriented unit triangle with vertices in S. Assume, in order to obtain a contradiction, that xa is the rightmost left edge with respect to x and yb is the leftmost right edge with respect to y for some a [not =] y and b [not =] x.
	www.findarticles.com /p/articles/mi_qa3742/is_200305/ai_n9186790 (1634 words)

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