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Circle–Point Midpoint Theorem

cirptmid.html

Exploration

"cirptmid_1.gif"

Graphics saved as "cir16.eps".

Show that the locus of midpoints from a fixed point "cirptmid_2.gif" to a circle "cirptmid_3.gif" of radius "cirptmid_4.gif", is a circle of radius "cirptmid_5.gif".  Furthermore, show that the center point of the locus is the midpoint of the segment between "cirptmid_6.gif" and the center of "cirptmid_7.gif".

Approach

Without loss of generality, choose the point "cirptmid_8.gif" to be the origin and the circle "cirptmid_9.gif" to have center "cirptmid_10.gif". Construct the locus of midpoints and examine its form.

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

Construct the circle and the locus of points.

Clear[h1,r1,t];
C1=Circle2D[{h1,0},r1];
pts=Point2D[Point2D[0,0],Point2D[C1[t]]]

"cirptmid_11.gif"

This locus is clearly a circle of radius "cirptmid_12.gif" centered at "cirptmid_13.gif", which is the midpoint of the line segment from the point to the circle's center.

Discussion

Here's a function that computes the midpoint circle in the special position.

Circle2D[Circle2D[{h_,0},r_]]:=
   Circle2D[{h/2,0},r/2];

The first plot is a numerical example with the origin outside the circle ("cirptmid_14.gif"), while the second plot's origin is inside the circle ("cirptmid_15.gif").

Map[(p0=Point2D[0,0];
     p1=Point2D[C1[Pi/6]];
     l1=Segment2D[p0,Point2D[C1[Pi/6]]];
     C2=Circle2D[C1];
     P=Point2D[p0,p1];
     Print[Sketch2D[{C1,C2,p0,p1,l1,P} /. #]])&,
    {{h1->2,r1->1}, {h1->2,r1->3}}];

"cirptmid_16.gif"

"cirptmid_17.gif"

Graphics saved as "cirptm01.eps".

Graphics saved as "cirptm02.eps".


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