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DELANGES TRISECTRIX AND SECTRIX
Curve studied by Delanges in 1783. 

Polar equation: .
Cartesian parametrization: (t = q / 2). Cartesian equation: . Rational circular quartic. 
Given a circle (C) (here, the circle with centre O and radius 2a) and a line (D_{0}) passing through the centre of the circle (here Ox), the Delanges trisectrix is the locus of the point M on a variable line (D) passing through O such that the line parallel to D passing through M cuts (C) at N in such a way that (ON) is a bisector of (D_{0}) and (D). 

Construction equivalent to the previous one: the circle (C) has radius a, A is a fixed point of (C), N is a variable point. The symmetry of the line (OA) with respect to (ON) intersects the tangent to the circle at N at point M .  
The Delanges trisectrix is the locus of the orthocentre
of a triangle with a fixed side the opposite vertex of which describes
a circle centred on the middle of the side, with radius the length of the
side multiplied by .
See a similar construction for the bicorn, the right strophoid, and the Kappa. 

The Delanges trisectrix is a special case of Cotes'
spiral.
The construction opposite shows the property of trisection: the angle MOP is the third of AOP. 

Its inverse curve with respect to O is the Dürer folium, which is, therefore, also a trisectrix.
Furthermore, the same construction shows that the curve
with equation
is an (n + 1)sectrix, that can be called "Delanges sectrix".
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© Robert FERRÉOL 2024