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CEVA TRISECTRIX AND SECTRIX
Curve studied by Ceva in 1699.
Giovanni Ceva (16481734): Italian mathematician and engineer. 
Polar equation: .
Cartesian parametrization: . Complex parametrization: . Cartesian equation: . Rational sextic. 
Given a circle (C) with centre O and radius a and a line (D) passing by O ((D) is here Ox), the Ceva trisectrix is the locus of the point M such that OP = PQ = QM with P on (C), Q on (D) and such that O, P and M are aligned.
The angle xOM is the third of the angle xQM, hence the name of trisectrix. Compare to the construction of the Maclaurin trisectrix. 


Like all tritrochoid, the Ceva trisectrix is the locus of the gravity centre of 3 circular motions. 

This curve is also a conchoid of the quatrefoil
(therefore, a conchoid of a rose).
The construction above can be continued, as shown in the figure below:
The curve of order n, with polar equation , is a (2n+1)sectrix, and is called "Ceva sectrix". 

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© Robert FERRÉOL 2017