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<Worksheet><Version major="6" minor="0"/><View-Properties><Zoom percentage="100"/></View-Properties><Styles><Layout alignment="left" bullet="none" firstindent="0.0" leftmargin="0.0" linebreak="space" name="Normal" rightmargin="0.0" spaceabove="0.0" spacebelow="0.0"/><Layout alignment="centred" bullet="none" name="Maple Plot"/><Layout alignment="centred" bullet="none" linespacing="0.5" name="Maple Output"/><Font background="[0,0,0]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" name="Text" opaque="false" size="12" underline="false"/><Font background="[0,0,0]" family="Times New Roman" foreground="[0,0,255]" name="2D Output" opaque="false" readonly="true" size="12"/><Font background="[0,0,0]" bold="true" executable="true" family="Monospaced" foreground="[255,0,0]" name="Maple Input" opaque="false" size="12"/></Styles><Group><Input><Text-field layout="Normal" style="Text"><Font size="14">Let's take two bars of length 9. Anchor the end of one bar at C=(-9.25, .25) and the end of the other bar at D=(9.25, -.25).  Observe that C and D lie on the line y=(-1/37)x.  Let's join their free ends with a bar of length 7/8.  Finally, let A=(a1,b1) and B=(a2,b2) denote the points at which the small bar attaches to the longer bars, and let P be the midpoint of the small bar.  The five equations that model this situation follow below.  Note that we obtain f1 and f2 come from the fact that A and B are constrained to lie on the circles of radius 9 centered at (-9.25, .25) and (9.25, -.25).  The distance formula for the length of the short bar gives us f3, and f4 and f5 give the location of P.   </Font>
</Text-field><Text-field layout="Normal" prompt="&gt; " style="Maple Input">f1:=(a1+37/4)^2+(b1-1/4)^2-81;</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation>NiM+SSNmMUc2IiwoKiQsJkkjYTFHRiUiIiIjIiNQIiIlRioiIiNGKiokLCZJI2IxR0YlRiojISIiRi1GKkYuRiohIyIpRio=</Equation></Text-field></Output></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">f2:=(a2-37/4)^2+(b2+1/4)^2-81;
</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation>NiM+SSNmMkc2IiwoKiQsJkkjYTJHRiUiIiIjISNQIiIlRioiIiNGKiokLCZJI2IyR0YlRiojRipGLUYqRi5GKiEjIilGKg==</Equation></Text-field></Output></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">f3:=(a2-a1)^2+(b1-b2)^2 -(7/8)^2;</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation>NiM+SSNmM0c2IiwoKiQsJkkjYTJHRiUiIiJJI2ExR0YlISIiIiIjRioqJCwmSSNiMUdGJUYqSSNiMkdGJUYsRi1GKiMhI1wiI2tGKg==</Equation></Text-field></Output></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">f4:=2*(x-a1)-(a2-a1);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation>NiM+SSNmNEc2IiwoSSJ4R0YlIiIjSSNhMUdGJSEiIkkjYTJHRiVGKg==</Equation></Text-field></Output></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">f5:=2*(y-b1)-(b2-b1);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation>NiM+SSNmNUc2IiwoSSJ5R0YlIiIjSSNiMUdGJSEiIkkjYjJHRiVGKg==</Equation></Text-field></Output></Group><Group><Input><Text-field layout="Normal" style="Text"><Font size="14">We would like to find the curve traced out by P as we move C into all its possible positions.  Note that because the joining bar AB is so short, we will not be able to go around the entire circle as in the previous example.  Also, since C and D lie on the line y=(-1/37)x, not the x-axis, P is going to move symmetric about the line perpendicular to y=(-1/37)x through the origin (0,0), which is the line y=37x.</Font></Text-field><Text-field layout="Normal" style="Text"/><Text-field layout="Normal" style="Text"><Font size="14">We now use resultants to eliminate all variables except x and y from these five equations.</Font></Text-field><Text-field layout="Normal" style="Text"/><Text-field layout="Normal" prompt="&gt; " style="Maple Input">res1:= resultant(f1,f4,a1);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation>NiM+SSVyZXMxRzYiLDIjIiNQIiIpIiIiKiRJI2IxR0YlIiIjRipGLCMhIiJGLUkieEdGJUYoKiRGMEYtIiIlKiZGMEYqSSNhMkdGJUYqISIlRjQjISNQRi0qJEY0Ri1GKg==</Equation></Text-field></Output></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">res2:=resultant(res1,f5,b1);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation>NiM+SSVyZXMyRzYiLDgjIiNQIiIpIiIiSSJ4R0YlRigqJEYrIiIjIiIlKiZGK0YqSSNhMkdGJUYqISIlRjAjISNQRi0qJEYwRi1GKkkieUdGJSEiIiokRjVGLUYuKiZGNUYqSSNiMkdGJUYqRjFGOSNGKkYtKiRGOUYtRio=</Equation></Text-field></Output></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">res3:=resultant(res2,f2,a2);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation>NiM+SSVyZXMzRzYiLDwqJEkieEdGJSIiIyElJkgiKiZGKEYpSSNiMkdGJUYpIiM7KiZGKEYpRiwiIiIiIikqJEkieUdGJUYpRi8qJkYoRilGMkYvISIpKiZGKEYpRjJGKSIjSyomRjJGKUYsRi9GMComRjIiIiRGLEYvISNLKiZGMkYpRixGKUYtKihGKEYpRjJGL0YsRi9GOiokRigiIiVGLSokRjJGOUY0KiRGMkY+Ri0=</Equation></Text-field></Output></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">res4:= resultant(f3,f4,a1);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation>NiM+SSVyZXM0RzYiLDAqJEkjYTJHRiUiIiMiIiUjISNcIiNrIiIiKiRJI2IxR0YlRilGLiomRjBGLkkjYjJHRiVGLiEiIyokRjJGKUYuKiZJInhHRiVGLkYoRi4hIikqJEY2RilGKg==</Equation></Text-field></Output></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">res5:=resultant(res4,f2,a2);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation>NiM+SSVyZXM1RzYiLFRJI2IyR0YlIyIlTDciIztJInhHRiUjISYmKj4lRioqJEkjYjFHRiUiIiMjIiZ2RCUiI0sqJEYrRjAjIiZWXCUiIikqJkYvIiIiRidGOSMhJnZEJUYqKiRGJ0YwIyImTnclRjMqJkYrRjlGJ0YwISRTKComRitGOUYnRjkhJCdIKiZGK0YwRidGMCIjUyomRitGMEYnRjlGKiokRisiIiQhJCNmKiRGKyIiJUYqKiZGL0ZHRidGOSEiJSomRi9GMEYnRjAhIiMqJkYvRjBGK0YwRjcqJkYvRjBGJ0Y5RkwqJkYvRjlGJ0ZHIiM3KiZGL0Y5RidGMEY3KiZGK0Y5Ri9GMCEkWyIqJEYvRkpGOSokRidGSiIiKiokRidGR0ZSIyEoJipHeCMiJSc0JUY5KihGL0Y5RidGOUYrRjAhIzsqKEYrRjlGL0Y5RidGOSIkJ0g=</Equation></Text-field></Output></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">res6:=resultant(res5,f5,b1);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation>NiM+SSVyZXM2RzYiLE4jISgmKkd4IyIlJzQlIiIiSSNiMkdGJSMiJUw3IiM7SSJ4R0YlIyEmJio+JUYuKiRGLyIiIyMiJlZcJSIiKSokSSJ5R0YlRjMjIiZ2RCVGNiomRjhGKkYrRiojISZ2RCUiIiUqJEYrRjMiJSFbJiomRi9GKkYrRjMhJSU9IiomRi9GKkYrRiohJCdIKiZGL0YqRjhGMyEkI2YqJkYvRjNGOEYzIiNLKiZGOEYzRitGKiEjOyomRjgiIiRGK0YqISNrKiZGOEYzRitGMyIjayomRi9GM0YrRjNGTyomRi9GM0YrRipGLiooRi9GM0Y4RipGK0YqRk0qKEYvRipGOEYqRitGKiIlJT0iKiRGL0ZMRkYqJEYvRj5GLiokRjhGPkYuKiZGOEYqRitGM0ZI</Equation></Text-field></Output></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">res7:=resultant(res6,res3,b2);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation>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</Equation></Text-field></Output></Group><Group><Input><Text-field layout="Normal" style="Text"><Font size="14">Notice res7 is a polynomial in just x and y.  We obtain the irreducible factors to this long polynomial by the following:</Font></Text-field><Text-field layout="Normal" style="Text"/><Text-field layout="Normal" prompt="&gt; " style="Maple Input">h:=factors(res7);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation>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</Equation></Text-field></Output></Group><Group><Input><Text-field layout="Normal" style="Text"><Font size="14">Since the first factor is a constant, we only need to second part of h, which we can simply obtain by the following command: </Font></Text-field><Text-field layout="Normal" prompt="&gt; " style="Maple Input">hh:=op(2,h);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation>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</Equation></Text-field></Output></Group><Group><Input><Text-field layout="Normal" style="Text"><Font size="14">The following command allows us to load implicitplot and use it to graph.</Font></Text-field><Text-field layout="Normal" style="Text"/><Text-field layout="Normal" prompt="&gt; " style="Maple Input">with(plots,implicitplot);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation>NiM3I0ktaW1wbGljaXRwbG90RzYi</Equation></Text-field></Output></Group><Group><Input><Text-field layout="Normal" style="Text"><Font size="14">We can plot the two components of hh as follows:</Font></Text-field><Text-field layout="Normal" prompt="&gt; " style="Maple Input">implicitplot(op(1,hh), x=-1.. 1, y=-2..2, numpoints=200000,view=[-1..1, -2..2]);</Text-field></Input><Output><Text-field layout="Maple Plot"><Plot height="400" type="two-dimensional" 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