Secant Angles in a Circle

Secant is another name for a transversal. It's a straight line that crosses another shape of interest. Secants that cross a circle have many different properties. Below, our concern is with the angles formed by two secants that meet in a point, say, A. Such angles are appropriately called secant angles. With the exception of the case where A lies on the circle, the secant angle intercepts two arcs on the circle: BC and DE, in the applet. The secant angle equals half the sum of the angular measure of the two arcs (for A inside the circle) or half their difference (for A outside the circle). In the case where one of the secants is tangent to the circle, A still lies outside the circle and the angle intercepts two arcs. The angle is half their difference, as before.

The exceptional case, where A lies on the circle, happens to be the most fundamental. It is treated separately and underlies the proofs for the other cases.

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Add one ancillary line to the diagram: DC. This creates a triangle ACD with an external angle BDC, for A outside the circle, and external angle BAC, for A inside the circle. In both cases, the inscribed angles BDC and DCE subtend the intercepted arcs. The result then follows from the Exterior Angle Theorem:

A is outside the circle      A is inside the circle
∠A= ∠BDC - ∠DCE
 = (arc(BDC) - arc(DCE))/2
∠A= ∠BDC + ∠DCE
 = (arc(BDC) + arc(DCE))/2

With the notion of the directed angle, the two cases reduce to one. When A is outside the circle, the arcs BC and DE have different orientations, and the corresponding inscribed angles have different signs. For A inside the circle, the signs are the same.

A special case where, say, AB is tangent to the circle, so that B and D coalesce, while C lies opposite to B across the circle, deserves special attention. In this case, ΔABC, by construction, while ΔBEC is right, because angle BEC subtends a diameter of the circle. The angles ABE (formed by a tangent and a secant at B) and BCE are equal, which allows us to conclude that the former has the same measure as the arc BE enclosed by its sides: the tangent AB and the secant BE.

(A different approach appears elsewhere.)

Angles in Circle

  1. Angle Subtended by a Diameter
  2. Inscribed Angles
  3. Inscribed and Central Angles in a Circle
  4. Munching on Inscribed Angles
  5. Sangaku with Angle between a Tangent and a Chord
  6. Secant Angles in a Circle
  7. Secant Angles in a Circle II
  8. Thales' theorem

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