Putting the Power of a Point Theorem to Work

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 display 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 (21.3 cm2) remains fixed.

Does this construction show us every possible rectangle with an area of 21.3 cm2? 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 cm2. (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 = AC2 = 21.3 cm2. 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 = AC2 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.”

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