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I'm Up and Down, and Round and Round

A corner standing on a diameter is a right angle

Nothing new is proved here. We put one number into the doubling rule: 180°.

A diameter AB, straight through the centre.
Step 1 of 4
  1. A diameter AB, straight through the centre.
  2. A point C, anywhere else on the circle.
  3. Triangle ABC, joined up.
  4. The corner at C, marked exactly 90°.

The rule

A diameter is a straight line through the centre. So it makes 180° at the centre. By the doubling rule, the edge angle is half of that. Half of 180° is 90°. Take a diameter AB and any other point C on the circle. Then angle ACB is exactly 90°, a right angle.

Exactly, not nearly

The 180° is not rounded. A diameter passes through the centre by definition. So the 90° is exact too.

Two near misses and the exact case, with the rim angle each gives
Angle at the centreAngle at the rimRight angle?
176°88°No
180° — a diameter90°Yes
184°92°No

A check with side lengths

In a triangle, the widest corner is opposite the longest side. Here the corner at C is a right angle. The other two corners total 90°, so each is smaller. Then the diameter opposite C is the longest side.

Move C round the circle. Then make AB shorter.

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This track

  1. 0 Start here
  2. 1 Four names for four parts of a circle
  3. 2 The line from the centre that cuts a chord in half
  4. 3 Same distance, same length — and further out is shorter
  5. 4 An arc makes twice the angle at the centre that it makes at the edge
  6. 5 Every point on one arc sees the chord the same way
  7. 6 Checkpoint: parts, chords and the doubling
  8. 7 A corner standing on a diameter is a right angle
  9. 8 Four corners on one circle, and the pairs that total 180°
  10. 9 Testing whether four points really share one circle
  11. 10 Three points give one circle, and the triangle shows where its centre is
  12. 11 Checkpoint: diameters, cyclic shapes and finding a centre
  13. 12 Many circles fit through two points
  14. 13 A circle looks the same when turned
  15. 14 Equal angles at the centre give equal chords
  16. 15 Equal angles on one side put four points on a circle
  17. 16 The big picture
  18. 17 Track quiz

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