The Swallowtail Model: Two Tails, One Simple Ratio

Step by step

1. Looks like college math? In China: grade school
   This looks like a college problem. In China, it's a grade school trick.
2. Point inside → three corners → two tails
   Join a point inside to all three corners. You get two tails.
3. 2 and 3. Left tail 12. Right tail?
   This line cuts the bottom into two and three. The left tail is twelve. How big is the right tail?
4. In China: "the swallowtail model"
   In China, this is called the swallowtail model, because the shape looks like a swallow's tail.
5. Both tails share this base
   Both tails share this line as their base. It runs from the top corner to the point.
6. Same base → compare the heights
   Same base, so their areas compare like their heights. A height is how far each corner is from this line.
7. 2 steps vs 3 steps → heights 2 : 3 → tails 2 : 3
   Now walk along the bottom, away from this line. Every equal step takes you the same extra distance away. The left corner is two steps away, and the right corner is three. So the heights are two to three, and so are the tails!
8. 2 shares = 12 → 1 share = 6 → 3 shares = 18
   So the left tail is two shares, and the right tail is three. Two shares are twelve, so one share is six, and three shares are eighteen!
9. Way 2: slide the point
   Here's a second way to see it. Let's slide the point along this line.
10. Same heights. Only the base changes.
    The corners don't move, so the two heights stay the same. Only the shared base changes.
11. Both tails change together → same ratio
    Cut the base in half, and both tails get cut in half. Double it, and both tails double. So the tails always compare the same way.
12. At the bottom: tails = the two parts
    Now slide the point all the way down. The tails become the two parts of the big triangle.
13. Same height, bases 2 and 3 → 2 : 3
    These share the top corner, so they have the same height. That's our bird's head tool. Their bases are two and three, so they compare two to three.
14. Ratio never changed → 2 : 3 → 18 again
    The ratio never changed while we slid. So our tails are two to three as well, and the right tail is eighteen again!
15. Tails compare like the bottom parts
    That's the swallowtail model. The two tails compare just like the two parts of the bottom.