Outline Mathematics
Geometry

Angles Subtended by a Diameter

Inscribed angles subtended by a diameter are right.

Angle On Diameter

Now see how this works:

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This is a known and a very useful property of inscribed angles that they measure half the central angle subtended by the same arc, or, which is the same, by the same chord. When a chord is a diameter, the central angles measures two,two,three,four right angles and the corresponding inscribed angles are all right,acute,right,obtuse. The applet above specifically demonstrates this fact, which is sufficiently important to warrant an independent proof. (The statement is often referred to as Thales' theorem.)

Let P be a point on a circle with diameter AB and center O. So that OA = OB = OP, as three radii of the same circle. This makes triangles AOP and BOP isosceles,isosceles,scalene,equal. In each, the base angles are equal and their sum equals the opposite exterior angle:

  ∠OAP + ∠APO = ∠BOP,AOB,APO,BOP,AOP and also
∠OBP + ∠BPO = ∠AOP,AOB,APO,BOP,AOP.

But since ∠OAP = ∠APO,AOB,APO,BOP,AOP and ∠OBP = ∠BPO, we further have

  2∠APO = ∠BOP,AOB,APO,BOP,AOP and also
2∠BPO = ∠AOP,AOB,APO,BOP,AOP.

Adding the two up

  2∠APO,AOB,APO,BOP,AOP + 2∠BPO = ∠AOP + ∠BOP,AOB,APO,BOP,AOP = 180°.

In other words,

  ∠APB = ∠APO,AOB,APO,BOP,AOP + ∠BPO = 90°.

Q.E.D.


Related material
Read more...

Angles in Circle

  • Inscribed Angles
  • Inscribed and Central Angles in a Circle
  • Munching on Inscribed Angles
  • Sangaku with Angle between a Tangent and a Chord
  • Secant Angles in a Circle
  • Secant Angles in a Circle II
  • Thales' theorem
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