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Auxiliary bars

Author:
Rafael Losada Liste
Topic:
Algebra, Congruence, Constructions, Geometry, Quadratic Equations, Geometric Transformations
This activity belongs to the GeoGebra book Linkages. If we want to prevent degenerate cases from appearing, we should add more restrictions to the system, if possible. For example, in the previous four-bar construction, if we want to avoid cases E' and F', where one pair of bars overlaps the other pair, we can add the black bar that appears in the following construction. Geometrically, this bar forces the EF bar to remain horizontal at all times. We can check this fact algebraically. Taking O=(0, 0) and U=(1, 0), the above four-bar system is given by the equations:
  • Ex2 + Ey2 = 1
  • (Fx - 1)2 + Fy2 = 1
  • (Fx - Ex)2 + (Fy - Ey)2 = 1
If we add to these equations the one corresponding to the black bar:
  • ((Ex+Fx)/2-1/2)2 + ((Ey+Fy)/2)2 = 1
then, a simple simplification leads us to the equalities:
  • Fx - Ex = 1
  • Fy - Ey = 0
which represent the horizontality of the bar EF, something that cannot be deduced from the previous three equations alone.
Author of the construction of GeoGebra: Rafael Losada

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