Thales' Theorem
Part of the Circles section of Coddy's Geometry journey. Lesson 32 of 62.
Take a diameter AB and any other point P on the circle. Join P to both ends of the diameter. The angle at P is always exactly 90°. This is Thales' theorem, one of the oldest results in geometry, and the reason is the inscribed angle rule.
The angle at P stands on the arc AB, and its central angle is ∠AOB. But A, O and B lie on one straight line, so ∠AOB is a straight angle, 180°. Half of 180° is 90°. The rule is called Angle in a semicircle: any angle at the circle standing on a diameter is a right angle.
Challenge
AB is a diameter of circle O. Drag the point P around the circle and watch the angle at P. Then press Check.
Try it yourself
Drag the figure and watch
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Circles
1Parts of a Circle
Center and RadiusChords and DiametersAll Radii Are EqualDiameter Is Two RadiiRecap: Parts of a Circle4Arcs and Sectors
Arc LengthArea of a SectorFind the Central AngleThe Missing Central AngleRecap: Arcs and Sectors7Angles in a Semicircle
Thales' TheoremThe Right Angle InsideIs That Enough?Semicircle With LettersRecap: Semicircles2Circumference
Around the CircleUsing the DiameterRadius From CircumferenceCircumference With LettersRecap: Circumference5Radii Make Isosceles
Two Radii, One TriangleBase Angles From the CenterThe Angle at the CenterIsosceles With LettersRecap: Radii and Triangles8Angles on the Same Arc
Same Arc, Same AngleUsing the Same ArcSpot the Bow TieSame Arc With LettersRecap: Same Arc3Area of a Circle
The Area FormulaArea From the DiameterRadius From the AreaCircumference to AreaRecap: Area of a Circle6Inscribed Angles
Meet the Inscribed AngleHalf the Central AngleTwice the Inscribed AngleThe Big ArcInscribed With LettersRecap: Inscribed Angles