Exploring Analyic Geometry with Mathematica®

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Equal Areas Point

eqarea.html

Exploration

Given ΔABC with vertices "eqarea_1.gif", "eqarea_2.gif" and "eqarea_3.gif" show that there are four positions of a point "eqarea_4.gif" such that ΔAPB, ΔAPC and ΔBPC have equal areas. The coordinates of "eqarea_5.gif" are given by

"eqarea_6.gif"

"eqarea_7.gif" is the centroid of ΔABC and "eqarea_8.gif". ΔABC connects the midpoints of the sides of "eqarea_9.gif".

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}

"eqarea_10.gif"

Solve the system of equations.

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

"eqarea_11.gif"

Construct points at the solutions.

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

"eqarea_12.gif"

Show that "eqarea_13.gif" is the centroid of ΔABC and "eqarea_14.gif".

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

"eqarea_15.gif"

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

"eqarea_16.gif"

Show that ΔABC connects the midpoints of the sides of "eqarea_17.gif".

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

"eqarea_18.gif"

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}]

"eqarea_19.gif"

Graphics saved as "eqarea01.eps".


Copyright © 1999-2007 Donald L. Vossler, Descarta2D Publishing
www.Descarta2D.com