The angle made by two
secants
intersecting outside a circle is half the difference between the
intercepted arc
measures.

Try this
In the figure below, drag the orange dots to reposition the secants. Note how the angles are related.
(Note: The angles are rounded off to whole numbers for clarity).

See also Intersecting Secant Lengths Theorem.

When two secants intersect outside a circle, there are three angle measures involved:

- The angle made where they intersect (angle APB above)
- The angle made by the intercepted arc CD
- The angle made by the intercepted arc AB

Recall that the measure of an arc is the angle it makes at the center of the circle. To see this more clearly, click on "show central angles" in the diagram above. For more on this see Angle measure of an arc.

The theorem still holds if one or both secants is a tangent. In the figure above, drag point C to the right until it meets A. The top line is now a tangent to the circle, and points A and C are in the same location. But the theorem still holds using the measures of the arcs CD and AB in the same way as before.

Make both lines into tangents in this way, and convince yourself the theorem still works.

- In the figure above click on "reset" then "hide details"
- Drag the points C and D to new locations
- Calculate the angle P
- Click on "show details" to check your result

- Circle definition
- Radius of a circle
- Diameter of a circle
- Circumference of a circle
- Parts of a circle (diagram)
- Semicircle definition
- Tangent
- Secant
- Chord
- Intersecting chords theorem
- Intersecting secant lengths theorem
- Intersecting secant angles theorem
- Area of a circle
- Concentric circles
- Annulus
- Area of an annulus
- Sector of a circle
- Area of a circle sector
- Segment of a circle
- Area of a circle segment (given central angle)
- Area of a circle segment (given segment height)

- Basic Equation of a Circle (Center at origin)
- General Equation of a Circle (Center anywhere)
- Parametric Equation of a Circle

- Arc
- Arc length
- Arc angle measure
- Adjacent arcs
- Major/minor arcs
- Intercepted Arc
- Sector of a circle
- Radius of an arc or segment, given height/width
- Sagitta - height of an arc or segment

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All rights reserved