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.

Morley triangle derived from the tripling of an angle

Morley equilateral triangle at the intersection of the trisectors
(from Wikimedia Commons)

In a previous post, I mentioned Morley's miracle: an equilateral triangle appearing at the points of intersection of the angle trisectors of any triangle.

Frank Morley was an excellent teacher and chess player (see some more about him at Pat'sBlog). If he had lived today, I like to think of him having his own blog communicating about his passion for math and chess, submitting his recreational problems. If he hadn't discovered the equilateral triangle at the intersection of trisectors, it's probable that this theorem would still be unknown. Trisection of angles is a controversial subject for research, especially among academia, where it is associated with suspicion of crankiness (see Underwood Dudley's book about trisectors). A pity... because trying to understand how to divide angles is a "natural" question, of which we shouldn't be ashamed, provided that we modestly take into account what has been found by other people.

All the proofs of Morley's theorem that I know of start with a result: an equilateral triangle or a triangle already drawn with its trisectors (see the different proofs referenced on the very complete French site Abracadabri or at the end of Alexander Bogolmony's Cut the Knot site). In this sense, they are backward proofs, which keeps some mystery about the physical origin of the equilateral triangle. I tried something different, a bit similar in spirit to my Archimedes tripling circle: How can we multiply an angle by 3, from which would result a Morley triangle embedded in a triangle of any shape? So here follows an alternative forward proof for Morley's theorem.

Step 1: I start with a circle in which I inscribe an angle α inferior to 60°. The lines are intercepting an arc on the circle.

Step 2: I duplicate the angle by tracing a little circle centered at one of the endpoints of the arc and with radius the chord of that arc. I repeat the same operation to triplicate the angle (see Figure 1).

Step 3: When two circles of equal size and common radius intersect, two equilateral triangles appear, so I draw both of them: Figure 2.

Step 4: I draw supplementary equilateral triangles at both sides, with the help of the intersections of the circles and the outer lines of the triplicate angle, see Figure 3.

Step 5: The magenta colored equilateral triangle is the Morley triangle of the arbitrary triangle we are looking for. The Morley triangle is a pivotal triangle, so the other vertices of the arbitrary triangle can be found through symmetric construction of the initial inscribing circle centered towards the other sides of the Morley triangle, see the red circles at Figure 4.

Step 6: I complete the figure with the triplicated angles β and γ in the red inscribing circles. The sides of the searched triangle are the outer lines of the triplicated angles, see Figure 5.

With this tripling angle construction, we always obtain an equilateral triangle at the intersection of the trisectors of a triangle. When the initial vertex is moved on the initial inscribing circle, all triangle shapes can be generated for any initial angle α between 0 and 60°, which proves the Morley theorem for any triangle.

Archimedes angle trisection or tripling?

Pi-day today, so a nice occasion to write something about circles. π is the ratio between the circles' perimeter (περίμετρος in Greek) and diameter.

In the previous post, I explored some angular tricks using circles. The circle is a key geometrical object when dealing with angles. For example, Archimedes' construction to trisect an angle uses a circle and a straight line with a marked unit. But for clarity's sake, I prefer to draw a set of circles which show how a straight line can mark odd multiples of a unit angle α on intersecting circles, see Figure 1. In that way, it is easier to see how Archimedes' trisection circle relates to the multiplication and division of angles by odd integers, see alternative Figure 2.

This makes me think that Archimedes' angle trisection is in fact the inverse of "Archimedes' angle tripling".

Playing with angles

A few months ago, I prepared this post but didn't feel ready to publish it. It raised more questions than it answered, so I wanted to have a little more background about it. Hopeless wish! There's so much about angles that I don't fully understand. With pi-day approaching, I'm also looking for some inspiration about circles and angles, so I thought publishing might get me in the direction...

I really became aware of the difficulty to divide an angle into three equal angles (a.k.a. angle trisection) some three years ago. It's bloody hard to find general procedures to divide precisely a circular arc into three equal arcs. There are some tricks that are well documented on the web, but you won’t find a simple geometric procedure using only a compass and an unmarked ruler. It has even been "proven" impossible. We only know of some "cheating" tricks like using a marked ruler (for Archimedes' trisection), origami or mechanical tools that will allow you to trisect precisely an angle. And there is an equilateral triangle that appears "by miracle" when you trisect all angles of a triangle, a result found by mathematician Frank Morley (because he himself tried to better understand or crack that impossibility?). So the angle trisection really became to intrigue me and I progressively dug into it.

One of the questions that has obsessed me is : How do we characterize an “Angle”? I’ve already written a post on that question, pointing out the utility of the chord subtending the circular arc of a circle of unit diameter. The measure of this chord is what we call the sine of the subtended angle. It characterizes the angle and it greatly facilitates our way to count angles, for example by progressing stepwise with chords in a circle.

Roughly speaking there are two ways to manipulate angles with circles. Either you use central angles, meaning that you fix the vertex and you radiate towards the circle in order to subtend an arc, or you use inscribed angles, meaning that you put the vertex of the angle on the circle. Figure 1 shows the equivalence between both methods. The central angle of a subtended arc in a circle of radius 1 is equal to the same angle when it is inscribed in a circle of diameter 1. This equivalence between central and inscribed angles is another way of stating that the measure of a central angle is exactly twice the measure of the inscribed angle, the factor 2 expressing the ratio between diameter and radius.

Figure 2 and 3 illustrate some advantages of inscribed angles. You can put the vertex anywhere on the circle: if the subtended circular arc (or its chord) remains constant in length, the angle remains the same. For the case when the vertex is located between the two endpoints of the subtended arc, you have to add the parts at each side of the vertex.

This flexibility of inscribed angles opens possibilities that remain hidden for central angles. If you combine different circles at intersections with straight lines, you’re simply calculating with angles. Figure 4 illustrates an alternative way to count angles. Starting with an angle α inscribed in a circle of unit diameter, if we draw a unit circle symmetric to the chord of the subtended angle, we can read off an angle (α+α) at the other side of the circle. Adding another circle results in an angle (2α+α) at the far side, etc. When the span of the angle exceeds the height of the unit circle, we can use extra circles in order to take over the continuation of the subtended arc (see the angles > 6α on Figure 4. So this simple procedure allows us to read off integer multiples of any angle.

The interesting thing is that wherever we shift a unit circle in this setting, the opposite subtended arcs differ always by an angle amount α, as shown in Figure 5 (in fact the circle needn't be aligned on the axis). So this allows us to add or subtract any angle to another angle.