Circle theorems: Thales
- Thales. The easiest circle theorem.
- Inscribed angle. The angle subtending a major arc is constant.
- Cyclic quadrilateral - opposite angles sum to 180°. Plus minor arc.
- Tangent radius. Blindingly obvious, but a slippery proof.
- Alternate segment. Perhaps the most unintuitive.
- Chord bisection by radius. The relevant angle is 90°.
- Equal length tangents. For completeness.
Thales's theorem is that the angle made between points A and B on the ends of a circle diameter and a third point P on the circle circumference is 90°.
Thales interactive
In the diagram below you can move point P on the circumference.
You can see that ∠APB is always 90°.
Thales: the proof
We first draw a line from O to P.
Now consider ΔOAP:
- |OP| = |OA| as both are radii.
- So ΔOAP is isosceles.
- Set the two identical angles to α.
Similarly ΔOBP is isosceles. Set its identical angles to β.
Sum ΔAPB angles: 180° = α + (α + β) + β which simplifies to α + β ∠APB = 90°. ∎
Where to now?
What if the chord AB is not a diameter? It turns out that ∠APB is still constant.
Many proofs are flawed: they consider only the simplest case. For more on this see the inscribed angle theorem.