DELANGES TRISECTRIX AND SECTRIX

 Curve studied by Delanges in 1783.

 The dotted curve is the Dürer folium, the inverse of which is the Delanges trisectrix. 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 (D0) 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 (D0) 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".