The power of a point theorem is one of the more surprising results in elementary geometry. The theorem says that if two chords *AB* and *CD* of a circle intersect at point *P,* then the product *AP · PB* is equal to the product *CP · PD*. You can see an illustration of this theorem in the Web Sketchpad model below. Drag points *A* and *C* and observe the measurements and calculations.

I was reminded of this theorem while reading the article Using Appropriate Tools Strategically by Milan Sherman and Charity Cayton in the November 2015 issue of NCTM’s *Mathematics Teacher*. Their article addresses the pedagogical potential of Dynamic Geometry technology and focuses on the power of a point theorem as a concrete example of how the software can foster high cognitive demand tasks.

Rather than summarize Sherman and Cayton’s ideas here, I’ll describe a Dynamic Geometry application of the power of a point theorem not covered in their article:* Use the power of a point theorem to build a constant-area rectangle. A constant-area rectangle is one whose perimeter can change but whose area remains fixed.*

Since the product of *AP *and* PB *is constant as point* A *travels around the circumference of the circle, this suggests that we construct a rectangle whose dimensions are* AP *and

*PB.*Press the arrow in the lower-right corner of the websketch above to move to the second page of the websketch, which displays this rectangle. Pressing

*Animate Point A*sends point

*A*around the circle while simultaneously changing the dimensions of the rectangle. While the perimeter of the rectangle changes, the power of a point theorem guarantees us that its area (30.32 cm

^{2}) remains fixed.

Does this construction show us every possible rectangle with an area of 21.3 cm^{2}? No. As point *A* spins, we don’t see rectangles where *AP’*s length is close to zero (like *AP* = 0.001 cm) and we don’t see rectangles where *AP’*s length is very large (like *AP* = 300 cm). Indeed, *AP* can be no larger than the diameter of the circle.

Might dragging point *P* very near the circumference work? Try it—you’ll notice that the constant area of the rectangle changes, and that clashes with our goal: We want to see all possible rectangles with an area of 21.3 cm^{2}. (Dragging point *P* outside the circle has interesting results; results that are beyond the scope of this post.)

Fortunately, there is another method of generating a constant-area rectangle. On page 3 of the websketch, you’ll see a right triangle *BDA*. Altitude *AC* is the geometric mean of *BC* and *CD,* meaning that *BC · CD* = *AC*^{2} = 21.3 cm^{2}. By animating point *C* along its horizontal line, distance *AC* remains constant, so we again create segments whose product is constant. But this time, as point *C* moves to the left, distance *BC* approaches 0 and distance *CD* approaches infinity.

There’s a nice connection to be made between these two methods of creating constant-area rectangles. Page 4 of the websketch again shows right triangle *BDA*, but now we see its circumcircle as well. Segments *AA’* and *BD* are chords of the circle with *BC* *· CD = AC · A’C, or * *BC · CD* = *AC*^{2} since *AC* = *A’C*. Thus the right triangle construction is really just a special case of the circle construction, with one of the two chords being a diameter of the circle.

I first wrote about constant-area rectangles in the April 1996 issue of NCTM’s *Mathematics Teacher*. Sketchpad has come a long way in twenty years, but my article’s conclusion still holds: “A chief pleasure of these investigations comes from taking theorems that may seem like nothing more than geometric curiosities and turning them into devices that perform a desired function. Specifically, the geometry that lies behind the chord theorem and the geometric-mean construction becomes the engine driving the movement of the constant-area rectangles. By setting these theorems in motion, students are able to generalize them and uncover relationships that the static counterparts in a textbook cannot reveal.”

Don’t stop here, look at the locus of the unlabelled point in picture 2.

My geometrical construction software/program/application/…/ is now working as a web page, using javascript

I would like it if you were to have a go with it.

Here is the link:

http://www.mathcomesalive.com/geostruct/geostructforbrowser1.html