CA2217809A1 - Thooem (two-um) - Google Patents

Abstract

(see fig. I) The double arc bisect method got put into a tangible form in the angle bisecting tool. Basically it is composed of two disks hinged together at their circumferences.

(see fig. II) Similarly the triple adjustment trisect method can be put into a tangible form in the thooem. Basically it is posed of two disks. The one has a nipple at its center.
The other is hinged to a central arm, and hinged to a chordal arm which has a slot for the nipple and a slot for alignment purposes.

Description

Description First draw a circle, angle, and bisect arcs. Place disk with holes over circle. The needles find the points; the buttons thread the needles positioning disk; gets clamped;
unthreaded; insert pegs; attach central and chordal arms. The nipple finds the groove;
rotate central, slide nipple, rotate, slide, mark the spot. Do other side. Accessories aid operation. Special parts for trisecting very small angles are to 1/6 scale; rest to 1/3;
except details to full.

March 17th 1997 * special thanks to business analyst Mary ~llen Heidt of S.I.D.C.O.
(Southern lnterior Development Cor-poration of the Okanagan-~imilkameen) in helpir-g to reduce description of the ~anipulation of the device to less than 75 words.

~pecifications radius = 15 centimeters material - solid wood model ~ T-1000 series or T-1001 Thooem Models T-1001 solid material (as sho~m) T-2001 clear see-through (not shown) T-3001 clear see-through (not shown) ~-1001 prototype (as shown) T-1002 working model (not shown) M.W.M. November 26th 1997 Geometry The tools of fundamental geometry or basic construction are well known to be a simple compass and an unmarked straight-edge.
With the simple tools it is possible to draw a random angle, and then divide it into two equal angles. ~he method of construction goes as follows:
1) - compass a circle 2) - strai~ht-edge a radius ~) - str3ight-edge another radius to form a random angle 4) - compass arcs from those two points on the circumference to find a point of inter-section which is the bisect point 5) - connect the bisect point to the center point of the circle or vertex of the angle with the bisect line ~ CA 02217809 1997-12-08 iv ii O ,~ iii ~ , The double arc bisect method is absolutely precise. The proof begins by connecting the bisect point to the two points on the circumference. This forms an equilateral quadrilateral because all four sides are radii.

~ CA 02217809 1997-12-08 iv 4 ~iii At the same time two triangle are formed which are exactly the same via the S.S.S.
triangle congruency principle. Therefor angle ii-i-iv is the same as angle iv-i-iii.
The double arc bisect mehhod can be proven in another way also, because an equilateral quadrilateral is a parallelogram which ha~,the property of ppposite angles being equal. Therefor the two triangles are the same via the S.A.S
triangle congruency principle.
It is also possible to prove something else!

V i ii ~.~ ~
"_ We have already proven that angle "y"
equals angle r'y'l. Now when we connect points ii-iii with a chord, the bisect line intersects the chord at point v. The triangles i-ii-v and i-iii-v are exactly the same via the S.A.S.
triangle congruency principle. ~herefor line segments ii-v and v-iii are equaI to each other!
This fact can be used if the construction were to continue.

This construction is long and complicated but all points are clearly defined. ~he method:
1) compass a circle 2) str~ight-edge a radius 3) another radius forms a random angle 4) connect those two points on the circumference with the chord of an isosceles triangle 5) put the double arc bisect method into play and the bisect line intersects the chord at a c~rtain point, the initial adjustment point 6) compass an arc from it finds two points on the bisect arcs 7) connecting those two points to the vertex of the angle finds two new points on the chord, the second adjustment points 8) compassi~g arcs from each of them finds two new points on the bisect arcs further out 9) connecting them to the vertex of the angle finds two new points on the chord, the third set of adjustment points 10) compassing arcs from each of them finds two new points on the bisect arcs a wee bit further out 11~ connecting them to the vertex of the angle apparently trisects the angle according to a measurement with a protractor 12) a geometrical proof follows /

Most of this has already been proven.
The bisector of the chord can also be proven to be perpindi cular to it. But this is not the feautre we build upon.

'I'hose two isosceles triangles can be shown to be exactly the sa~e thanks to S.S.~.

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~o then angle z e~uals angle z And T.~e already have angle x equal to x And there is the axiom 'l~he sums of equals added to equals are equal. !l tS"

So those two angles are equal And those two triargles congruent via S.A.S.

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So then the two outside angles have been shown to be the same, and this carries on ~C

I ~ A. ~ . A. a t~ri a-~- gle c ongruency principle ?

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If so.

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~ s And w~ got to go through that all again.

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1~ 1 ll And through it again.

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~o get to here, where those two angles have been conclusively show~ to be e~ual for sure.

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If we could prove that the center angle is e,ual to one of the outside angles that would prove that all three are equal.
~o just consider, if you will, the distance from point xiv to point xviii, or the following dotted line ~' f/

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e _, ,Ji' .. ........

If that dotted line is a radius, then we would have an equilateral ~uadrilateral, ~hich has a property of a diagonal seperating it into two congruent triangles, which would prove that the ce:l~ter angle is the same as the one on the outside.

This is ;ot a blank page. Above is a drawing of a circle, angle, iso cles triangle, double arc bisect & triple adjustment with a double pointed compass. The sa.rne compass mea~ures a certain two points to be a radius apart.

Analt~sis A compass and unmarked str-,ight-edge construction can be plotted on the Gartesian coordiaate system with (x,~) points and lines in the y=mx + b format and circles in the x2 + y2 = 1 format. ~ince there is an old and well known meth~d to trisect a right angle of 90~ we can figure out the slope of a 30~ angle.

m = .57735~

Now we can run the new method through the equations to see if the third adjustment fou~d the slope of a ~0 angle.

m = ,579876 It came up a few thousandths of a slope in the ~0~ range short. Therefor the triple adjustment trisect method is not absolutely precise. Then one wonders if another adjustment would find it, but after the fourth we get m=.577844 which is still a lit.le bit short.

After the fifth adaustment the slope m=,577447 which is still short. After the sixth adjustment the slope m=.577369 which is still short. After the seventh adjustment the slope m=.577354 which is still short but shows that the rnethod is bee bopping towards the ~0~ line, or rather the method is honing on in on it.

It takes an infi~ite number of adjustments to trisect an angle precisely. ~owever a grid can only be made so small, only so man~ dots per inch. Meanwhile the rlew method of const-ruction can be programmed to infinity with simple algabraic instructions.

2~

Perhaps the new method of con~truct-io~ could be put into a computer program or graphic displav device or plotter, but that is beside the point of this letter.
T~!iih just a simple compass and unmarked straight-edge an~ angle can be trisected according to a naked eye observation, and the thooem could never hope to accomplish more than that.

3o ~ ~ top view j~ , .~ .

, ~ . . , r -~ 5 ide view _ _ L I _ The double arc bisect mehhod got put into a tangible form in the angle bisecting tool. Basically it is composed of two disks hinged together at their circumferences.

. .

~imilarly the triple adjustment trisect method can be put into a tangible form in the thooem. Basically it is com-posed of two disks. The one has a nipple at its center.
The other is hinged to a central arm, and hinged to a chordal arm which has a slot for the nipple a~d a slot for alignment purposes.

Claims

Claims Page #44 is the claims.

They will make sense when the completion occurs.

The thooem patent being sought is very simple, but the device comes in three different forms. The one has the long central arm and gets made out of opaque material like oak for instance. The second has the short central arm and gets made out of clear stuff like acrylic for instance. The third has no central arm whatsoever, and it gets made out of clear stuff as well.

The thooem shown in the drawings has buttons for spacing purposes, to aid the manipulation, but i don't plan on making that contraption. The buttons aren't necessary if the thooem gets designed a little differently. All three models are very simple. You'll see when you get the drawings.