Shoulder Point on Median

camedian.html

Exploration

Let C be a conic arc with control points , and and projective discriminant ρ. Let P be the point on the median associated with vertex of such that ( is the midpoint of ). Show that P is coincident with the shoulder point of C, having coordinates

.

Approach

Construct the geometry and compare the coordinates of P to the shoulder point coordinates.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Create the conic arc control points.

Clear[x0,y0,xA,yA,x1,y1];
p0=Point2D[P0={x0,y0}];
pA=Point2D[PA={xA,yA}];
p1=Point2D[P1={x1,y1}];

Construct the midpoint of the chord.

pM=Point2D[p0,p1]

Construct the point on the median.

Clear[p];
P=Point2D[pM,pA,p*Distance2D[pM,pA]] //Simplify

Construct the shoulder point.

Clear[xM,yM];
Q=Point2D[{
   xM + p (xA - xM) /. xM->(x0+x1)/2,
   yM + p (yA - yM) /. yM->(y0+y1)/2}] //Simplify

The point on the median is coincident with the shoulder point.

IsCoincident2D[P,Q]

Discussion

This is a plot of a numerical example.

ca1=ConicArc2D[P0,PA,P1,p];
Sketch2D[{ca1,p0,pA,p1,pM,Q,
          Segment2D[pM,pA]} //. {
   x0->0, y0->0, xA->2, yA->6, x1->6, y1->0, p->0.65},
   PlotRange->All]

Graphics saved as "camedi01.eps".