Equal Areas Point

eqarea.html

Exploration

Given ΔABC with vertices , and show that there are four positions of a point such that ΔAPB, ΔAPC and ΔBPC have equal areas. The coordinates of are given by

is the centroid of ΔABC and . ΔABC connects the midpoints of the sides of .

Approach

Construct the geometry and solve a system of equations that equates the areas of the three triangles.  Compare the centroid and midpoints as specified.

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 points.

Clear[xA,yA,xB,yB,xC,yC,x,y];
A1=Point2D[xA,yA];
B1=Point2D[xB,yB];
C1=Point2D[xC,yC];
P=Point2D[x,y];

Compute the areas of the triangles.

a1=Area2D[Triangle2D[A1,P,B1]];
a2=Area2D[Triangle2D[A1,P,C1]];
a3=Area2D[Triangle2D[B1,P,C1]];

Form equations by equating the areas (squared).  Squaring is required because the area calculation may produce a symbolically negative number for the area.

{eq1=a1^2==a2^2,eq2=a2^2==a3^2}

Solve the system of equations.

ans=Solve[{eq1,eq2}, {x,y}]

Construct points at the solutions.

{P3,P2,P1,P0}=Map[(Point2D[x,y] /. #)&,
                  ans];
{P0,P1,P2,P3}

Show that is the centroid of ΔABC and .

Point2D[Triangle2D[A1,B1,C1],Centroid2D]

Point2D[Triangle2D[P1,P2,P3],Centroid2D]

Show that ΔABC connects the midpoints of the sides of .

{Point2D[P1,P2],Point2D[P1,P3],Point2D[P2,P3]}

Discussion

This is a plot of a numerical example.

Sketch2D[{Triangle2D[A1,B1,C1],
          Triangle2D[P1,P2,P3],
          P0,P1,P2,P3} /.
   {xA->-1,yA->-1,
    xB->3,yB->0,
    xC->1,yC->3}]

Graphics saved as "eqarea01.eps".