In the quiet elegance of geometric construction, sometimes a simple configuration reveals a profound truth. This post explores one such configuration, part of a legacy left behind by my late Vietnamese friend, Nguyen Tan Tai, who referred to it as the Lam-Ca construction — from the Vietnamese words meaning “do it all.” And that’s exactly what these diagrams aim to do: to explain, prove, and illuminate all at once.
What Nguyen Tan Tai created is more than a diagram: it is a visual proof that brings together trigonometric identities, reciprocal relationships, and the power of constant area.
Click the image or open the interactive Geogebra diagram here.
Unit Diameter Circle Instead of Unit Radius
Most trigonometric constructions begin with a circle with radius 1, where sines and cosines are represented as vertical and horizontal projections of a point on the circle. But in this configuration, the circle has unit diameter.
In a circle of unit diameter:
The angle α is inscribed.
The opposite chord has a length equal to sin α.
For an inscribed right triangle (i.e. with hypotenuse = 1) The adjacent chord has a length equal to cos α.
The projection of the opposite and adjacent chords on the hypotenuse have respective lengths sin²α and cos²α.
We therefore use chord lengths derived directly from the geometry of the circle. They reflect the sine and cosine values as segments formed within the circle, preserving symmetry and proportions naturally. The squares (blue and red) and rectangles (green and orange) of the upper part of the figure illustrate this nicely. I have already posted this construction in a previous post.
The Insightful Twist: Sliding Rectangles of Constant Area
What Nguyen Tan Tai introduced was an original twist on the classical representation of the inscribed rectangle: instead of constructing squares on the chord segments, he formed rectangles (shown in grey in the lower part of the figure) defined by:
Left rectangle: one side is sin α, the other is 1/sin α
Right rectangle: one side is cos α, the other is 1/cos α
The result? Each rectangle has a constant area of exactly 1, even as the angle α changes. In this way we see a whole family of unit-area rectangles. Their lower vertices trace a horizontal line anchored at y = –1. The shape of the rectangle dynamically adapts, while its area remains invariant.
Why This Matters
This construction transforms reciprocal trigonometric identities from abstract formulas into tangible, visual objects. The construction reveals them as side lengths in a dynamic geometry.
This approach:
Makes reciprocal relationships intuitive
Shows how geometry can express trigonometry with visual clarity
Bridges sine and cosine with their multiplicative inverses
Highlights a rarely explored link between circular geometry and constant-area transformations
The Lam-Ca Perspective
What makes the Lam-Ca construction special is its simplicity: no coordinates, no algebra, just pure geometry. Using circles of unit diameter and basic projections, Nguyen Tan Tai’s approach recovers deep identities from nothing but straightedge-and-compass logic.
This is not just a visual trick. It’s a philosophy: that geometry, when done right, can do it all.
A Quiet Tribute
To our knowledge, this method of constructing unit-area rectangles with sides defined by chord-based trigonometric values and their reciprocals has not been formally documented elsewhere. It stands as a tribute to Nguyen Tan Tai’s creativity and insight—a quietly powerful expression of geometry’s poetic depth.
This visualization is a piece of his legacy, and I’m honored to share it. What if some of the most beautiful trigonometric identities could be seen rather than derived algebraically?
In the next post, I’ll extend this same diagram to reveal tripling angle identities like cos(3α)
and sin(3α)
— all through sliding rectangles and simple symmetry. Stay tuned.