Angles Formed Outside a Circle

When two lines meet at a point outside a circle — two tangents, two secants, or one of each — the angle between them is set by the two arcs they cut off. That angle is exactly half the difference of those arcs.

Theorem Half the difference of the arcs

If two secants, two tangents, or a secant and a tangent meet outside a circle, the angle formed equals half the difference of the two intercepted arcs.

The far arc is BD and the near arc is AC, so the angle at E is:

\alpha = \frac{\overset{\frown}{BD} - \overset{\frown}{AC}}{2}

Note Three configurations, one rule

The same rule covers three cases: two tangents, a tangent and a secant, or two secants — all meeting at a point outside the circle.

A tangent touches the circle at one point and is perpendicular to the radius there; a secant cuts the circle at two points. Whichever pair you have, take the far arc, subtract the near arc, and halve.

Example One from each case

Two tangents

The two arcs are 250° and 110° (they add to 360°).

\alpha = \dfrac{250^\circ - 110^\circ}{2} = \dfrac{140^\circ}{2}

⟹ α = 70°.

Tangent and secant

The far arc is 120° and the near arc is 40°.

\alpha = \dfrac{120^\circ - 40^\circ}{2} = \dfrac{80^\circ}{2}

⟹ α = 40°.

Two secants

The far arc is 100° and the near arc is 40°.

\alpha = \dfrac{100^\circ - 40^\circ}{2} = \dfrac{60^\circ}{2}

⟹ α = 30°.

Reference Outside, inside, and on the circle

Lines	Vertex	Angle
Two tangents	outside	(far − near) ÷ 2
Tangent & secant	outside	(far − near) ÷ 2
Two secants	outside	(far − near) ÷ 2
Two chords	inside	(arc₁ + arc₂) ÷ 2
Tangent–chord	on the circle	arc ÷ 2

Summary

Two lines meeting outside a circle make an angle equal to half the difference of the two intercepted arcs.
The rule is the same for two tangents, a tangent and a secant, or two secants.
Outside the circle you subtract the arcs; when two chords cross inside, you add them.
Memory aid: vertex outside the circle → subtract; vertex inside → add.