 Given a circle with a diameter of 120 parts, we can use elementary geometry to find several different chord lengths, notably the chord on the arc of 60 degrees, the chord on the arc of 72 degrees, the chord on the arc of 90 degrees, the chord on the arc of 120 degrees. But what about other chords? To find these other chords, we can use a theorem proven by Ptolemy and the Alma Guest, now known as Ptolemy's theorem. Let quadrilateral A, B, C, D, B inscribed in a circle, then the rectangle on the diagonals A, C, and B, D is equal to the rectangle on A, B, C, D, together with the rectangle A, D, B, C. We can read this in terms of the sums of products of lengths, so the product of the diagonals is equal to the sum of the products of the opposite sides, or A, C, times B, D is equal to A, B, times C, D, plus A, D, times B, C. Now while this result is true in general, a useful special case, and the one Ptolemy focuses on, occurs when one side is the diameter. So let's try to find the chord of 30 degrees. The key to doing this is we want to express 30 degrees in terms of chords we already know. Of course it helps to know what chords we already know, so remember Ptolemy found the chord of 60, 72, 90, and 120 degrees. So we note that 30 degrees is 90 degrees minus 60 degrees, and so we begin with the chords of 90 degrees and 60 degrees. Now we could have used 30 degrees equals 120 degrees minus 90 degrees, but the chord of 60 degrees is a whole number, so it's easier to work with. Let's draw some pictures. If we let A, D, B the diameter, and A, B the chord of 60, A, C the chord of 90, then B, C will be the chord of the difference 30 degrees. Since A, B is the chord of 60 degrees, we know that length. A, C is the chord of 90 degrees, so that's going to have a length 84, 51, 10. Now since A, C is the chord of 90 degrees, then C, D will also be the chord of 90 degrees, and so that will also be 84, 51, 10. And then B, D, since A, B is the chord of 60, that means B, D is going to be the chord of 120, so that'll be 103, 55, 23. And we can use Ptolemy's theorem, the product of the diagonals is equal to the sum of the products of the opposite sides. And so filling in those lengths. Now strictly speaking, we should work with the sexogasimal products. However, in deference to the fact that most of us are not used to working with sexogasimals, we'll convert to decimals and find the length of B, C. Although if we want to remain true to Ptolemy's actual work, we'll give a final answer in sexogasimal, 31, 330. For the chord of half the angle, Ptolemy provides a useful result. Let A, B, C, D be a quadrilateral inscribed in a semicircle with diameter A, D, where B, C is equal to C, D. If we drop C, F perpendicular to A, D, then D, F will be half the difference between A, D and A, B. To prove this, we'll let A, E equal A, B, and we'll join C, E. Then, well, you should be able to complete the rest of the proof on your own. The important things here are that A, B and A, E are equal, and that angle B, A, C and C, A, D subtend equal arcs. What this means is that we'll be able to get the length of this segment that's cut off by the perpendicular. And that's important because we also have the following. Let arc B, D be bisected at C and we'll drop C, F perpendicular to A, D. Then the rectangle on A, D and D, F is equal to the square on C, D. And this one is straightforward from similarity. We note that triangle A, C, D is actually similar to triangle C, F, D. And so there's a proportionality between the sides where C, D is a common side, and that gives us our rectangle on A, D and D, F equal the square on C, D. And the previous proposition told us how to find D, F, and A, D is just going to be the diameter. So for example, we can find the cord of 45 degrees. This is the cord of half of 90 degrees. So let's draw a picture. If we let A, D be the diameter, and C, D a cord of 45 degrees, B, C another cord of 45 degrees, then B, D and A, B are cords of 90 degrees. And we have the following. A, D, that's the diameter. And A, B is a cord of 90 degrees, and we already know that length. Now we've bisected the cord, so if we drop the perpendicular C, F, then Ptolemy's second theorem tells us that D, F is half the difference between A, D and A, B. And we know the lengths of A, D and A, B. So we can find D, F, 17, 34, 25. And now we know that the rectangle on A, D and D, F is the square on C, D. And we know all the lengths except for C, D. And again, working in decimals, we find a value for C, D. And to remain faithful to Ptolemy's work, we'll convert that back into sex decimals 45, 55, 19.