Limacon2

Pascal's Limacon, another generation

Consider the circle c₀ with diameter AB and a second circle c passing through {A,B}. For a point P varying on c₀ extend {PA,PB} to define the intersection points correspondingly {A',B'} with c. Then
[1] Triangles PAB and PB'A' are similar.
[2] Line PD is orthogonal to A'B', D being the middle of AB.
[3] Quadrangle DOFP is a parallelogram, F being the middle of A'B' and O the center of c.
[4] Line A'B' is tangent to the fixed circle c' with center O and radius equal to the radius of c₀.
[5] The locus of E is a limacon of Pascal (see Limacon.html for its definition).

Consider the circle c₀ with diameter AB and a second circle c passing through {A,B}. For a point P varying on c₀ extend {PA,PB} to define the intersection points correspondingly {A',B'} with c. Then
[1] Follows by the equality of angles at {A,A'} and {B,B'}.
[2] Is a consequence of [1] by measuring the angles at P.
[3] By [1] applied to PA'B', PF is orthogonal to AB. It follows that quadrangle DOFP has parallel opposite sides.
[4] Is a consequence of [3].
[5] Follows from the definition of the limacon (see Limacon.html ) and the previous properties.

Pascal's Limacon, another generation

See Also