Angles Formed by Secants and Tangents Outside a Circle

Picture standing outside a circle and aiming two lines into it. Each line crosses the circle and cuts off an arc on the far side from where you're standing, and the two lines together trap a near arc between them, closer to your vertex. The angle where the lines meet is not the average of these arcs the way an inscribed or inside angle would be. It's the gap between them: half of the far arc minus the near arc. The farther arc pulls the angle open, the nearer arc pulls it closed, and what's left over is the angle you actually see.

This single rule covers three setups that look different but behave the same way. Two secants crossing outside the circle, a secant and a tangent meeting outside, or two tangents drawn from the same outside point — in every case, identify the arc far from the vertex and the arc near the vertex, subtract, and halve. A tangent just touches the circle at one point instead of cutting through two, so its 'arc' is the single arc it grazes, but the subtraction logic doesn't change.

This sits right next to the rule for angles formed inside a circle, where two chords cross and the angle equals half the sum of the two arcs. Outside the circle, you subtract; inside, you add. Keeping that contrast straight is the real skill here, since the arc-finding and the halving are the easy parts.