Circle–Point Midpoint Theorem

cirptmid.html

Exploration

Graphics saved as "cir16.eps".

Show that the locus of midpoints from a fixed point to a circle of radius , is a circle of radius .  Furthermore, show that the center point of the locus is the midpoint of the segment between and the center of .

Approach

Without loss of generality, choose the point to be the origin and the circle to have center . 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]]]

This locus is clearly a circle of radius centered at , 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 (), while the second plot's origin is inside the circle ().

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}}];

Graphics saved as "cirptm01.eps".

Graphics saved as "cirptm02.eps".