Law of sines, chords and similar triangles
27 November 2016 | 9:29 am

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.
{\displaystyle {\frac {\sin A}{a}}\,=\,{\frac {\sin B}{b}}\,=\,{\frac {\sin C}{c}}}
while in this case, we should really write:
{\displaystyle {\frac {\sin A}{a}}\,=\,{\frac {\sin B}{b}}\,=\,{\frac {\sin C}{c}}\,=\,{\frac {\sin 90^{\circ }}{d}}}

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 C0B 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.

It From Bit or Bit From It? FQXi essay
30 March 2013 | 4:54 pm

2013 FQXi essay contest is announced. Topic "It From Bit or Bit From It?" This topic is somehow connected with the 2011 topic "Is Reality Digital or Analog?", not my favorite one, as I have grown my conviction that the it from physics is what really what underlies reality. Talking about "Bits" then just talking about data, information, sensations perceptions, formulations that originate from the "It"

I've been writing FQXi essays three times in row:
Never in the prizes, but enjoying the writing. This time, I think I'll pass my turn as I need to finish my PhD thesis before this summer, which enables me to apply some ideas of my 2009 essay to some unsolved experimental issues in semiconductor nanophysics. A question of focus.

For those who'll compete, enjoy and good luck!

Dreaming in Geneva - FQXi essay
4 September 2012 | 8:27 pm

The theme for this year's FQXi contest topic is "Questioning the Foundations: Which of Our Basic Physical Assumptions are Wrong?". I had some difficulty to start with this topic (I didn't seem to be the only one, see Ajit Jadhav's blog here and there). I had a lot of things to say about what has gone wrong with physics, which assumptions had to be reconsidered. So, since the opening of the contest, I regularly put some ideas in a draft, being confident that I would be able to arrange them into a coherent thesis for the essay. However by the 20th of August (ten days before closing), I still didn't know how I could write them together into an essay without being suspected of "trotting out my pet theory" (see warning in the Evaluation Criteria).

My "pet theory" is simple: the fundamental entity in physics is "THE quantum particle" which you can represent as an arrow (a vector, a ket). From the mechanical interactions between such rod-like particles, you may deduce all of physics, provided that you assume some complementary parameters (such as the velocity at which two particles fly one from another = c, the length of the rod = Bohr diameter of hydrogen). No mass, no force, no charge, etc. Just paths of rotating arrows that interact with each other through contact (collision). This is the way I reason about photons, electrons, quarks, fields, waves, etc. But I can't reasonably write it that way in an essay. I would need to recall a lot of history of science. So I chose to bring up some ideas that have emerged in history of science that we could reconsider, not necessarily in the same way, but gaining insight with hindsight.

Also I prefer to avoid abstract mathematics when talking physics. Mathematics is just a language, very convenient though, but really just a language that can hinder us in our intuitive understanding. Instead of math, scientists could as well use words, fantasy, dreams, pictures, poems maybe. It is an art and sometimes it is necessary to change the expression of this art. I hope you'll enjoy my dreaming in Geneva.


More News from this Feed See Full Web Site