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Topic: Trisecting the angle

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	[No title] (Site not responding. Last check: 2007-11-05)
		In the construction of a nine-gon, it is necessary to trisect a 120 deg.(2pi/3) angle.
		Given an angle that has both sides intersecting a single circle at least once, the measure of the angle is 1/2 of the difference between the measures of the smaller central angle and the bigger central angle.
		Thus, angle CAB is the one to be trisected.
	www.perseus.tufts.edu /GreekScience/Students/Tim/Trisection.page.html (788 words)

	From the time of Euclid, geometric constructions were done solely with a straightedge and compass (Site not responding. Last check: 2007-11-05)
		and the angle AOB is trisected by OC.
		His method of trisection was that a circle is described with a center being O, and that center point being the vertex of an angle.
		The angle AOB, where O is the common vertex, has been trisected with the coordinates P. The Limacon invented by Pascal, later proved to be useful in trisecting an angle.
	www.cerebral-palsy.net /trisectangle/trisectangle2.html (1751 words)

	trisecting an angle
		Whereas bisecting an angle could hardly be simpler, splitting an angle in three equal parts with compasses and straightedge alone is impossible, except in a few special cases such as when the angle happens to be 90°.
		Trisecting an arbitrary angle can be done if you cheat by using a measuring ruler instead of a plain straightedge, or even if you draw just two marks on the straightedge, but not if you play by the rules and the straightedge is completely blank.
		In fact, the question of whether trisection could ever be done in the general case remained open until 1837, when it was finally shown to be impossible by Pierre Wantzel, a 23-year-old French mathematician.
	www.daviddarling.info /encyclopedia/T/trisecting_an_angle.html (249 words)

	Search Results for angle*
		Trisectors of an angle, squarers of the circle, discoverers of perpetual motion, constitute a class of mankind whose peculiarities deserve the attention of the student of human nature, and whose personal history is often calculated to awaken the deepest commiseration.
		A symmedian of a triangle from vertex A is obtained by reflecting the median from A in the bisector of the angle A. Durell
		a) The summit angles are > 90 degrees (hypothesis of the obtuse angle).
	www-history.mcs.st-and.ac.uk /Search/historysearch.cgi?SUGGESTION=angle*&CONTEXT=1 (8192 words)

	Trisecting the Angle
		Actually this proof is related to the proofs that trisecting the angle, doubling the cube and squaring the circle are impossible.
		It is possible to trisect the angle using a marked straightedge, but that's not allowed by the ancient Greek rules, since a mark can't be lined up against another point, line or arc without trial and error, and without some inherent error of alignment.
		The radius of the spiral is proportional to azimuth.
	www.uwgb.edu /dutchs/PSEUDOSC/trisect.HTM (4996 words)

	Trisecting the Angle
		Trisecting the angle is one of the three classic unsolved problems of antiquity.
		Thus angle AOB = 3EOB and EOB trisects AOB.
		Trisect chord AB, use CD for a first approximation at trisecting the arc, trisect the chord difference, and get a second approximation at trisecting the arc.
	www.uwgb.edu /dutchs/PSEUDOSC/Trisect0.HTM (585 words)

	Trisecting an Angle History Summary (Site not responding. Last check: 2007-11-05)
		Trisecting an angle is one of the three classical problems of Greek geometry, together with doubling the cube and squaring the circle.
		The problem of trisecting an angle has been proven to be impossible, but mathematicians daily receive "proofs" from amateur mathematicians who believe they have found a way to trisect the angle.
		Most of the people who believe they have found a way to trisect the angle have inadvertently used the straightedge for measuring; often their error is buried so deep that it is difficult to detect.
	www.bookrags.com /history/mathematics/trisecting-an-angle-wom (629 words)

	mathschallenge.net (Site not responding. Last check: 2007-11-05)
		Trisecting an angle using compass and straight edge is impossible.
		Hence the trisection of an angle is not possible by compass and straight edge.
		Despite the Greeks being unable to trisect an angle using compass and straightedge (which was proved, in 1847, by the method above), Archimedes knew of another method for trisecting the angle.
	mathschallenge.net /index.php?section=faq&ref=library/constructions/trisecting_angle (428 words)

	Trisecting an angle
		The problem of trisecting an arbitrary angle, which we examine here, is the one for which I [EFR] have been sent the largest number of false proofs during my career.
		Although it is difficult to give an accurate date as to when the problem of trisecting an angle first appeared, we do know that Hippocrates, who made the first major contribution to the problems of squaring a circle and doubling a cube, also studied the problem of trisecting an angle.
		Gauss had stated that the problems of doubling a cube and trisecting an angle could not be solved with ruler and compasses but he gave no proofs.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Trisecting_an_angle.html (1994 words)

	[No title]
		The trisection of an angle, or, more generally, dividing an angle into any number of equal parts, is a natural extension of the problem of the bisection of an angle, which was solved in ancient times.
		The spiral discovered by Archimedes was used for angle trisection and more generally to solve the problem of cutting an angle contained by straight lines in a given ratio.
		In his discussion of the trisection of an angle, he said, "Nicomedes trisected any rectilineal angle by means of the conchoidal curves, the construction, order and prope rties of which he handed down, being himself the discoverer of their peculiar character.
	www.math.rutgers.edu /courses/436/436-s00/Papers2000/jackter.html (2256 words)

	Math Forum - Ask Dr. Math
		Many people think they have found trisections, but they either don't understand exactly what the problem is or their method is flawed.
		An exact trisection in a finite number of steps is also required since it's easy to do a trisection that doesn't require too much accuracy.
		As a limiting case, it might be possible to perfectly trisect an angle in an infinite number of steps.
	mathforum.org /library/drmath/view/51706.html (564 words)

	Trisecting the angle (from geometry) -- Encyclopædia Britannica
		Trisecting the angle (from geometry) -- Encyclopædia Britannica
		The Angles gave their name to England, as well as to the word Englisc, used even by Saxon writers to denote their vernacular tongue.
		The Pythagorean Theorem is used to calculate the relationship between the legs and angles of a triangle.
	www.britannica.com /eb/article-217479 (871 words)

	Angle Trisection Problem: Trisect An Angle With a Tomahawk (Site not responding. Last check: 2007-11-05)
		Trisecting an angle is impossible with a straight-edge and compass, but a special tool called a 'tomahawk' makes this construction possible.
		With the tomahawk in this position, the angle is trisected by the "handle", and by the line (not shown) connecting the angle vertex to the center of the semi-circle.
		A geometric proof of why the angle is trisected is within the reach of high school geometry students, and makes a great conclusion to the project.
	www.articlesforeducators.com /math/000005.asp (660 words)

	: Angle Doubling, review at WorldSSP.net (Site not responding. Last check: 2007-11-05)
		Math: Running fixesThe som of angles in a triangle is 180°Let line DAE be parallel to line BC, then the angles a and b equal angles DAB and EAC, respectively.
		Trisecting an angleThere are three classical problems in Greek mathematics which were extremely influential in the development of geometry.
		DOUBLING THE ANGLE ON THE BOWRelative bearing can be eyeballed by ADDING the desired bearing (say 45°R) to the vessel’s compass course (say 105°C) and sight over the vessel’s compass for an object at 150°C. Note the time the object is sighted at the relative bearing.
	www.worldssp.net /webinfo_m.asp?proid=4003 (437 words)

	Angle Trisection
		I answered the second question by saying it was impossible to trisect an angle with a straightedge and a compass, and gave the person a reference to some modern algebra books as well as an article Evelyn Sander wrote about squaring the circle.
		When told that it is impossible to trisect an angle with a straightedge and compass people then often believe it is impossible to trisect an angle.
		First take the angle to be trisected, angle ABC, and construct a line parallel to BC at point A. Next use the compass to create a circle of radius AB centered at A. Now comes the part where the marked straightedge is used.
	www.geom.uiuc.edu /docs/forum/angtri (1278 words)

	Trisecting the Angle: Archimedes' Method -- Encyclopædia Britannica
		More results on "Trisecting the Angle: Archimedes' Method" when you join.
		Three approaches can be distinguished: Hippocrates' dodge of substituting one problem for another; the application of a mechanical instrument, as in Hippias's device for trisecting the angle; and the technique that proved the most fruitful, the...
		His device depends on a curve, now known as the quadratrix of Hippias, that is produced by plotting the intersection of two moving line segments, as shown in the.
	www.britannica.com /eb/article-9126483 (764 words)

	Three Impossible Constructions
		The three classical construction problems of antiquity are known as ``squaring the circle'', ``trisecting an angle'', and ``doubling a cube''.
		Given an angle, construct an angle whose measure is exactly 1/3 the measure of the original angle.
		The problem of trisecting an angle is similar, but what is usually done is to show that there is a particular angle that cannot be trisected, and that angle is typically chosen to be
	mathcircle.berkeley.edu /BMC3/construct/node12.html (437 words)

	Trisecting an angle
		The problem is therefore to trisect an arbitrary angle and the aim is to make the construction using ruler and compass (which is impossible) but failing that to devise some method to trisect an arbitrary angle.
		Later, however, they trisected the angle by means of the conics, using in the solution the verging described below.....
		angle PBA = angle POA (angle at centre of a circle is twice the angle at the circumference standing on the same arc).
	www-gap.dcs.st-and.ac.uk /~history/PrintHT/Trisecting_an_angle.html (1994 words)

	The Trisection of an Angle (Site not responding. Last check: 2007-11-05)
		It turns out that trisecting the angle is equivalent to solving a cubic equation.
		degrees cannot be solved by ruler and compass and thus the trisection of the angle is not possible.
		It is possible to trisect an angle using a compass and a ruler marked in 2 places.
	db.uwaterloo.ca /~alopez-o/math-faq/node57.html (229 words)

	Angle Trisection
		This is a simple straightedge and compass construction of a third part of an angle.
		Angle ODE is equal to angle OED, and their sum is equal to the sum of DOC and FOE.
		The angle DOC is exactly one third of the angle AOB.
	mentock.home.mindspring.com /trisect.htm (197 words)

	TRISECTING AN ANGLE
		Draw a circle with the angle in its centre and draw to radii's following the sides of the angle.
		If it does you subtract the length from 'y' and to the edge of the angle from 'd'(not mathematically though, you have to do it graphically by accommodating your compass to the space between the line 'y' and the intersection point on the yellow circle), we call this new length 'z'.
		Then draw the circle with the radii 'd' whose centre is at the edge of the angle and then draw 'l2', all as described in pic7.
	www.physics.us /gnu/angle.htm (885 words)

	The Harvard Crimson :: News :: Mechanical Means of Trisecting Angle Found
		Undaunted by the fact that mathematicians have slaved for centuries over the problem of trisecting an angle, Harvey "Scott" Sleeper '42, of Eliot House pondered about it in his spare time and has at last developed a mechanical means of graphically solving the problem.
		Sleeper had been told by his teachers that the trisection of an angle was one of those things that simply could not be done geometrically.
		The apparatus is a simple two-piece device which, when placed correctly over the angle, shows where the lines of trisection are to be drawn.
	www.thecrimson.com /article.aspx?ref=185799 (303 words)

	Mathematical Mysteries: Trisecting the Angle
		Trisecting an arbitrary angle can be done if you cheat by using a measuring ruler instead of a plain straight edge (you can find out how in the sci.math FAQ file), or even if you simply draw two little marks on your straight edge.
		The problem of whether trisection could be done in the general case remained a mathematical mystery for millennia - it was only in 1837 that it was eventually proved to be impossible by Pierre Wantzel, a French mathematician and expert on arithmetic.
		Pierre showed that the two problems of trisecting an angle and of solving a cubic equation are equivalent.
	plus.maths.org /issue7/xfile (651 words)

	Trisecting an angle (Site not responding. Last check: 2007-11-05)
		For example, since it is possible to construct a 30 degree angle, a right angle is trisectable.
		Therein it states that since general trisection is related to cubic equations, and only those related to equations that may be solved using square roots are generally constructable, we cannot trisect the angle or duplicate the cube.
		We know how to make divisions according to powers of 2, because such divisions are based upon the intersections of pairs of lines (or arcs, as the case may be).
	www.newton.dep.anl.gov /newton/askasci/1995/math/MATH101.HTM (316 words)

	69th AB - Abstract (Site not responding. Last check: 2007-11-05)
		Two of the three classical Greek problems, which cannot be solved by using ruler and compasses alone, are to trisect a general angle and to duplicate a cube.
		The problem of trisecting the angle (finding an angle the size of which is exactly one third of the size of a given one) can be solved e.g.
		This shows that a general angle can be trisected by using compasses and a "marked ruler", the same set of instruments as in the case of duplicating the cube.
	adela.karlin.mff.cuni.cz /~rokyta/api/doc/abstr069.htm (1197 words)

	Trisection Unraveled (Site not responding. Last check: 2007-11-05)
		Trisection of an Angle Unraveled (by Quadratic Equations) manuscript was developed over a period of thirty years, the main body prior to 1972.
		The indirect proof that a trisection cannot be constructed has been accepted by academia for over 150 years.
		Their angles, when trisected are of the highest importance, for they occupy a precise position of the squared circle and incorporate a distance of pi.
	www.trisectionunraveled.com /unraveled.htm (705 words)

	Introduction
		In the above diagram, we drew out the trisecting lines determined after only 3 steps, and we see that we have already trisected the angle, at least within the limits of the physical representation of angles on this page.
		We see that each pair of numbers is converging to the anticipated trisection values for Y = 75; namely, 25 and 50, for the respective angle values.
		A graphical technique, in the tradition of the old Greek masters, was described to trisect an angle.
	members.cox.net /topseeder/tritext2.htm (1072 words)

	Explorations in Math
		This particular segmentation of the diameter is clearly not a trisection of the diameter.
		When told that it is impossible to trisect an angle with a straightedge and compass, people often think it is impossible to trisect an angle.
		This concludes the explorations of trisecting an angle.
	jwilson.coe.uga.edu /emt669/Student.Folders/Godfrey.Paul/work/proj2/tri.html (1158 words)

	Trisecting an Angle Problem for MAT 429.31 (Site not responding. Last check: 2007-11-05)
		In the next section we will see how we can trisect an angle if we do not impose the limitations of using only compass and unmarked straightedge.
		Slide the straightedge on the diagram, keeping S on the line segment PR and T on the segment PQ, and thus locate H on the segment PR and K on the segment PQ to satisfy these conditions: HK = 2BP, and B, H, and K are collinear.
		The line segment BK is a trisector of the angle ABC.
	www.math.ilstu.edu /smccrone/TIP_2000/Trisecting.html (140 words)

	trisecting the angle
		That is, make an arbitrary angle and triple it by copying it twice using the well known construction for copying angles.
		Odds are you will find that your method requires you to trisect one angle in order to trisect the second.
		That is you can't trisect until you have already trisected - a contradiction.
	www.geocities.com /CapeCanaveral/Launchpad/5577/musings/trisect.html (331 words)

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