Angle in a Semicircle Is 90°: The Proof (Thales' Theorem)

Step by step

1. Any point on the circle → always exactly 90°
   Pick any point on this circle. This angle is always exactly ninety degrees.
2. Slide C: ∠A changes · ∠C stays 90°
   Slide C anywhere. The triangle changes shape, the angle at A changes, but the angle at C stays ninety.
3. AB goes through the center: a diameter. Why 90°?
   The secret is A B. It runs from edge to edge through the center: a diameter. But why ninety?
4. Step 1 · OA = OB = OC (radii)
   Step one. Draw a line from the center, O, to C. O A, O B and O C are all radii, so they're equal.
5. Step 2 · △AOC is isosceles → two angles a
   Step two. Triangle A O C has two equal sides, so it's isosceles, and its two base angles are equal. Call them a.
6. Step 3 · △BOC is isosceles → two angles b
   Step three. Triangle B O C is isosceles too, so its base angles are equal. Call them b.
7. Step 4 · 2a + 2b = 180° → a + b = 90°
   Step four. The big triangle has an a at A, a b at B, and a plus b at C. Together they make one eighty. So two a plus two b is one eighty, and a plus b is ninety.
8. ∠C = a + b = 90°, wherever C is
   And the angle at C is exactly a plus b. Ninety degrees, wherever C sits.
9. Thales' theorem · an ox… or was it Pythagoras?
   This is Thales' theorem. One ancient story says Thales was so proud of it that he sacrificed an ox. But another writer told the same story about Pythagoras.
10. Check it: a cup, a line through the middle, a paper corner
    Check it yourself. Trace around a cup, draw a line through the middle, and fit the corner of a sheet of paper on the circle. It fits, every time.
11. On a test: radii → isosceles ×2 → angle sum → 90°
    On a test: O A, O B and O C are radii, so they're equal. Triangle A O C is isosceles, so its base angles are both a. Triangle B O C is isosceles, so its base angles are both b. The angles of triangle A B C add to one eighty. So two a plus two b is one eighty, and a plus b is ninety. That's angle A C B.
12. On a test, it looks like this: find the radius
    On a test, it can hide like this. Find the radius.