Rotation Geometry: Spin It Any Way — These Two Lines Stay Equal

Step by step

1. Two equilateral triangles share a corner · spin the small one
   Two equilateral triangles share one corner. Spin the small one any way you like.
2. Pink line = green line, always. Why?
   Join the far corners with a pink line and a green line. They always stay exactly the same length. Why?
3. "The hand-in-hand model": they hold hands at the shared corner
   In China, this is called the hand-in-hand model: the two triangles hold hands at the shared corner.
4. Way 1: the pink triangle = big side + small side + pink line
   Way one: look at this pink triangle. It's made of a side of the big triangle, a side of the small one, and the pink line.
5. Turn it 60° around the shared corner
   Now turn it sixty degrees around the shared corner, and watch where it lands.
6. Big side → big side · small side → small side (60° apart)
   The big side lands on the next big side: same triangle, sixty degrees apart. The small side lands on the next small side, the same way.
7. Pink lands on green → equal, and they meet at 60°
   So the pink line lands exactly on the green line. They're equal! And they meet at sixty degrees, the turning angle.
8. Way 2: congruent triangles (SAS)
   Here's the school way: congruent triangles, side, angle, side.
9. Big side = big side · small side = small side
   These two sides are equal: both are sides of the big triangle. These two are equal: both are sides of the small one.
10. Each angle = 60° + the same shared angle → equal
    Now the angles between them. Each one is sixty degrees, plus the same shared angle in the middle. So they're equal too.
11. SAS → congruent → pink = green
    Side, angle, side: the two triangles are congruent. So the pink line equals the green line.
12. Hand in hand → the far-corner lines are equal and meet at 60°
    So when two equilateral triangles hold hands at a corner, the lines joining their far corners are equal, and meet at sixty degrees.
13. Turning keeps lengths the same → rotations in high school geometry
    Where will you see this? Whenever a shape turns around a point, its lengths stay the same. That's the idea of rotations in high school geometry.
14. On a test: CA = CB, CD = CE, ∠ACD = ∠BCE → SAS → AD = BE
    On a test, write the proof like this. Two pairs of equal sides. Equal angles: sixty plus the shared angle. Side, angle, side gives congruent triangles. So the two lines are equal.