Sliding Rectangles and the Lam-Ca Construction

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?

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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.

Unveiling Pythagoras: How Circle-Based Area Transformations Reveal the Truth

I recently had a discussion about the Pythagorean theorem and how to understand it better. I realized that the role of the circle as a fundamental element is often underrated. You can hardly find any proof where the right-angled triangle is placed within a circle to support the reasoning. Instead, the triangles are usually presented in a plane without reference. However, when you position the triangle inside a circle, it becomes much easier to visualize how the relationship holds as the right-angle vertex moves along the circle.

The key idea to understand is that in the following image, for a unit square, the green and blue areas are always equal to cos²(α), while the orange and red areas correspond to sin²(α). It is evident that the square, with an area of 1, is the sum of its left and right sections.

By opening the GeoGebra resource [Pythagoras proof in a circle (opens a new tab)], you can slide the right-angle vertex (red point) anywhere along the upper half of the circle. 

Law of sines, chords and similar triangles

Recently, I picked up my pencil and notebook and started to draw circles and triangles again. Swiftly, by drawing similar triangles circumscribed by circles, I got interested in their proportionalities, leading me to the law of sines.
For any triangle ABC, where a is the length of the side opposite to angle A, b the length of the side opposite to angle B and c the length of the side opposite to angle C, the law of sines states that:

where d is the diameter of the circle circumscribing ABC, as traced in Figure 1.

Figure 1: Arbitrary triangle ABC circumscribed in its circle with diameter d

My search in literature and web left me somewhat frustrated, for different reasons:

• one often omits to mention the diameter d, in its statement and even in its proof,
• one rarely develops this very elegant statement to closely related geometrical principles, like the intercept theorem or similarity transformations
• the historical background of the law of sines couldn't be checked easily from its original sources.

Why we should not leave out the diameter d of the circumscribed circle

The law is often stated in the reciprocal form and leaving out the diameter.

while in this case, we should really write:

We should have in mind that the sine of an angle is the length of the chord of the same angle inscribed in a circle of unit diameter, as you can teach children with spaghetti.

The law of sines can be understood as a statement relative to proportions between:
- chords traced in a circle with unit diameter,
- and similar chords traced in a circle scaled up by a factor d.

For example, we could trace a circle with unit diameter tangent at A inside the original circle, see Figure 2. With the notation used in this figure, the scale factor d can then be read out easily as different ratios: d = a/a₀ = b/b₀ = c/c₀.

Figure 2: Triangle ABC and tangent unit circle in A

This is one of the intuitions behind the law of sines. and we could view it as a natural law of similarity: "Corresponding lines in similar figures are in proportion, and corresponding angles in similar figures have the same measure."

Of course, if we want to set up a formal proof, we can deduce it from other laws :

1. The inscribed angle:

When inscribed in the same circle, all angles, subtending arcs of the same measure, are equal.

In Figure 2, BC being of the same length as DE, the angles  and are equal.

Therefore sin A = "opposite side over hypotenuse" = a/d.

Q.E.D.

This is also visually explained at "Better explained" or for those who read French "Blog de maths".

2. Central and inscribed angle, with Pythagoras:

This proof involves an additional notion: the central angle.
One can find it on other good sites:
Pat Ballew's blog
Math less travelled

3. Using the height of the triangle
This proof comes in different variations, either through expressing the area respective to the different heights (as given on wikipedia), either through expressing one height as ratios with two different sides (this is the academic proof, example here). Not my favorite one, as it doesn't give any insight in the scale factor d. If you don't need to pass exams, but doing math for fun, please forget this one!

Homothetic transformation with scale factor d

A homothety is a transformation where a geometric entity is transformed a similar version with a scale factor. In Figure 2, we represented the homothety from a circle of unit diameter towards a circle with diameter d. The inscribed lines, triangles and other polygons undergo the same scaling. And thus, we can complement the law of sines with a list of other ratios that also equal d.
In Figure 3, I draw the unit circle at an arbitrary place in space. Then joining similar points. The intersection of the lines is the homothetic center O.

Figure 3: Triangle ABC as a homothety from A₀B₀C₀ centered in O

Now, any line passing through O and intersecting with the small circle, will also intersect with the large circle at similar points (example D and D₀, C and C₀, B and B₀, etc.) The ratios of various line segments that are created if we trace pairs of parallels from these points will be the same, for example:
CD/C₀D₀ = BD/B₀D₀ = AD/A₀D₀ = OD/OD₀ = d
This is the intercept theorem.

Curiously, when searching on the web, both laws, the law of sines and the intercept theorem aren't often associated, while they are, in my opinion, stemming from the same basic principle of conservation of proportions.

I refer to two interesting posts that are related:
At Math is fun: Theorems about Similar Triangles
At Girls' Angle: Do you believe this?

Historical background

And in the history of geometry?

I've looked up sources about Apollonius of Perga, Ptolemy, Regiomontanus, Viete, Coignet (http://logica.ugent.be/albrecht/math/bosmans/R007.pdf), Simson (Elements of the conic sections), Jakob Steiner, but couldn't always find the original sources. I would be interested to have access to them.

The same for a paper by Richard Brandon Kershner. "The Law of Sines and Law of Cosines for Polygons." Mathematics Magazine, vol. 44, no. 3, 1971, pp. 150–153. www.jstor.org/stable/2688227.

Lost theorem about angular proportions

Last week, I came across a so called missing theorem about angular proportions in a triangle, discovered (or rediscovered) by Leon Romain. This triangle construction is presented by user Linelites on youtube.

For such a "Romain triangle", the missing theorem states that, if one of the inner angles is twice another inner angle, one has the property a² = bc +c², where a, b and c are outlined in Figure 1.

The sine chord pattern presented in the Teaching sine function with spaghetti provides helpful insights for the missing theorem. In this pattern, all angles at the intersection of chords are integer multiples of a chosen unit angle or its complement, modulo 90°. All segment lengths in this pattern can therefore be written as sums or differences of cosine and sine products and ratios of that angle. Figure 2 pictures some measures of sine chords in a circle of unit diameter, for an arbitrary angle θ.

Figure 3 shows a Romain triangle for angle α in this pattern. There are numerous other Romain triangles in this pattern. Can you figure them out? With the help of the measures pictured in Figure 2, one can follow visually the elements of a proof for the missing theorem.

Side b is the sine of the chosen angle θ (Figure 4):

b = sinθ

Side c times sin3θ equals b times sinθ (Figure 5):

c = sin²θ/sin3θ

Side a equals b cosθ minus c cos3θ (Figure 6)

a = sinθ cosθ - sin²θ cos3θ /sin3θ

Working out a² and bc +c², one finds that they are both equal to 2 sin²θ cosθ / sin3θ.

The interesting thing is that the sine chord pattern hosts plenty of "missing theorems" about angular proportions, which are only waiting to be (re)discovered.

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Update (June 7, 2010): I corrected Figure 6, which held a wrong term.

Teaching sine function with spaghetti

Sine functions appear everywhere in physics and mathematics. This seems to be related to the circular symmetry of space and to the periodic behavior of dynamical processes. The usual definition is that the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle (definition at wikipedia). The sine is also often represented as the ordinate of a point running on a unit circle centered at the origin. Any right triangle can be put in that setting. The length of the blue segment on Figure 1 illustrates this definition for an arbitrary angle α, in a circle of unit radius.

It is of interest to have in mind other geometric representations of the sine function. This helps to associate figures with sine and cosine operations or identities. The red segment in Figure 1 shows such an alternative representation, because the sine is also a chord of a circle of unit diameter. When we increment the angle by steps α, one endpoint of the chord advances on the little circle, while the other endpoint remains fixed at the origin, as illustrated by the animation in Figure 2.

When it represents the sine of angle α, the chord in the little circle spans an angle 2α. It is therefore a direct way to the angle multiplication and division by 2. Moreover, the endpoints of this chord can be placed two by two in any other direction on this circle. This enables one to find other variations on the recurrence pattern of the angle in the circle. During one of my circle drawing sessions, I was surprised by the stepwise zigzagging pattern of Figure 3.

Teaching the sine function with this zigzag pattern is particularly suited for daily life situations. Children like it, especially when you explain it at the table.