**Problem by Peter Ash** (geometry forum 12 April 2006)

Let P be a point on a circle with center O, and let Q be a point on ray OP,
with OQ > OP. Let R and S be the points where the tangents from Q intersect
the circle, and let T and U be the midpoints of the segments RQ and SQ. Find
the locus traced out by T and U, as Q moves.

The curve looks something like y = x^{4}. Does it have a name? Can anyone
find a simple form of the equation for the curve?

*Solution*

Equation is obtained using the locus equation feature of Cabri-Geometry II
Plus.

The figure is built like the Luigi Tomasi's example in geometry forum:

the circle has equation: *x*^{2}+*y*^{2}=1, and the
point P has coordinates (0; 1).

Click thumbnail image:

Calculating the equation (by François Rideau and Michel Tixier).

Choose parameter t so that R coordinates are R(sin(t);cos(t)). So Q( 0;1/cos(t))
and T( (1/2) sin(t);(1/2)( cos(t) + 1/cos(t)) )

Then elimination of t via sin(t)^{2} + cos(t)^{2} = 1 gives
cartesian equation 4*x*^{4} + 4*x*^{2}.*y*^{2
}- 4*x*^{2 }-* y*^{2} + 1 = 0

Alternate cartesian equation (by Michel Tixier):

y = (1-2x^{2}) / √(1 - 4 x^{2}) with x in ] -1/2 , 1/2
[.

x = ±1/2 are two asymptotes of the curve.

Changing radius r = OP for 1,2,3,4,... we can "deduce" with Cabri
the general formula:

4*x*^{4 }+ 4*x*^{2}.*y*^{2 }- 4r^{2}.*x*^{2
}- r^{2}.* y*^{2 }+ r^{4} = 0

Calculating equation as above confirms this result.

Up to our knowledge this curve has not got a special name.

Jean-Marie Laborde proposes a generalization : *what happens if you
choose another point than the midpoint, e.g. a point at a constant ratio on
QR* ?

Joseph Hormière found in the case r = 1 the equation x^{4} +
x^{2}.y^{2}– 2k.x^{2}– k^{2}.y^{2}
+ k^{2} = 0 where k is the ratio (k=1/2 gives Peter Ash's case).

François Rideau proposes another problem: *the construction of the
tangent to the quartic at point T* .

This problem is difficult from the mathematical point of view and the confirmation
with Cabri is amazing.

Here is the François Rideau's figure:

Figure (QuartiqueFR.fig) in French

Figure (QuarticFR.fig) in English

Other loci can be constructed thanks to F. Rideau's figure. The pedal of the nephroïd is a rose:

The pedal of the quartic is a non-regular quadrifolium (four-leafed rose):

Figure (PodaireJB.fig) in French

Figure (PedalJB.fig) in English

NB Downloading Cabri geometry figures : if your web navigator do not recognize Cabri files, you can use local popup menu for downloading (Windows: right button, MacOS : ctrl-button).