## Exploring Analyic Geometry with |
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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".

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