Geometry › Circle theorems
Circle theorems
When chords, tangents and inscribed triangles meet a circle, surprising relations always hold.
Circle theorems
Three classic results — pick one and watch the relationship hold.
Inscribed angle theorem
The inscribed angle is always half the central angle that subtends the same arc.
Inscribed angle theorem
An inscribed angle has its vertex on the circle and is formed by two chords. A central angle has its vertex at the center. If both subtend the same arc, the inscribed angle is exactly half the central angle.
Thales' theorem
A special case of the inscribed angle. If one side of an inscribed triangle is a diameter, the opposite angle is exactly 90°. Useful for constructing right angles with just a compass.
Tangent perpendicular to radius
A line that just touches the circle at one point — a tangent — is always perpendicular to the radius drawn to that point.
More circle results
• Angles in the same segment are equal.
• Opposite angles in a cyclic quadrilateral sum to 180°.
• The intersecting chords theorem: AB × CD = ... (chord segments multiply equally).
Why these matter
Circle theorems chain together. Once you know one angle in a circle problem, you can usually deduce every other angle and length using the rules above. It's the reason competition geometry problems often involve circles — they're puzzle-rich.