Exploring Circles

One of the treats for my 84th birthday just passed was to spend an hour with my brother trying to prove a conjecture, which I do not remember ever having been taught at school. 

(It recalled a similarly enjoyable hour spent 62 years earlier. David, Peter and I, were final-years students each reading for different degrees; David (Greats), Peter (PPP), and I (Botany). We found ourselves meeting regularly on Friday evenings at the "Welsh Pony", a now-vanished pub down by Gloucester Green, Oxford. Conversation roamed; on that occasion we were trying to remember the proof of Pythagoras' famous theorem involving the square on the hypotenuse.)

I had recently noticed that the diameter of a circle subtends an angle (at any point on the circumference) that looks very like 90°. Is it exactly 90°? Why were we never told about this at school? 

Figure 1
In Fig. 1, XZ is a diameter, Y is any point on the circumference, YW is a second diameter. Angle XYW is called a, WYZ is called b. The hypothesis is that angle XYZ (= a+b) = 90°

I showed this to my brother who became equally intrigued. We scribbled away for an hour, occasionally sharing the progress we had made. Success came the next morning. It seemed to us convincing that, by symmetry, we can label four radii as equal, four angles as equal to a, and four equalling b.  Furthermore, in the triangle YCZ, the angle sum (= 180°) is 2a + 2b; so a+b = 90°. Q.E.D. This would be true wherever we placed point Y.
After bathing in the glory of this success for an hour, I wondered if I could prolong this happiness by floating some more hypotheses. What, for example, if the chord was not a diameter but was shorter, as (for example) XW in Fig. 1. That chord clearly subtends angles at Y and Z that are less than 90°, but they are nevertheless patently equal. What is more, the chord XW subtends and angle at the centre that is twice that at the circumference. Of course they are special cases, as the lines XZ and WY are diameters. So I drew Fig. 2. (Below).

Figure 2
In Fig. 2, AB is a diameter, as is CD, passing throught the centre O. AO is therefore the radius, length r. AC is a chord of length r, thus subtending an angle of 60° at the centre and 30° at the circumference. CF and BF are also Chords of length r. 

Once again, AOB and COD are diameters (going through centre O). But the chord AC (likewise CF, FB, and BD) was chosen to be exactly r  because I know (from previous work) that exactly six such chords fit round a circle, forming exactly six equilateral triangles with their tips at the centre. It can be seen from Fig. 2 that chord AC subtends and angle of 60° at the centre and 30° at the circumference. 

In fact, all chords subtends two angles at the circumference. In general one is less 90°, and the other is greater than 90°, unless the chord is a diameter, when both are 90°.  

The chord CB subtends an acute angle (60°) at D, and an obtuse angle (120°?) at F. This example illustrates further property of chords.  The obtuse angle is supplementary to the acute angle (60° + 120° = 180°). For another example, the chord CF subtends an acute angle of (30°) at D, an angle (COF) of 60° at the centre (i.e. twice that at the circumference); and an obtuse angle of 150° at E  (supposing angle-sum quadrilateral is 360°, and you remember that FC and FO are both radii.)

References:

Wikipedia: Chords, Hipparchus,. 

There are many YouTube and other videos on Chords, Tangents and Secants

Apparently chords have been much studied since Babylonian times. The Greek geometer and astronomer, Hipparchus,  wrote a 12 volumed book on Chords around 150 BC, though that book itself is now lost. Tables of chords were used as we have used tables of sines.]