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 Skip Navigation LinksMath Help > Geometry > Polygons and Triangles > Intersecting Chords - intercepted arcs

The Central Angle Theorem can be extended to any pair of intersecting chords, not just those that happen to intersect on the circle.  This extension can be expressed this way:

Intersecting chords form an angle equal to the average of the arcs they intercept.

If the chords intersect inside the circle, and are subtended by arcs of x and y, then the angle formed by the intersection of the chords is (1/2) (x+y)

If the chords intersect outside the circle, and are subtended by arcs of x and y, with x > y, then the angle formed by the intersection of the chords is (1/2) (x-y)

You can think of these formulas as equivalent if you imagine the signed arc length being negative if the arc bows in toward the vertex of the angle formed by the intersecting chords, and the signed arc length being positive if it bows outwards.   In that case, the angle formed by the intersection of the chords is always (1/2) (x+y)

Proof

Proof of Intersecting Chords TheoremIn a triangle, an exterior angle is the sum of the other two interior angles.  By adding the red line in the illustration to the right, then from the Central Angle Theorem, we see that the two angles marked in red are x/2 and y/2, since they subtend arcs of x and y, respectively.  Thus, the exterior angle of this triangle is (x+y)/2, as marked.

Related pages in this website:

Summary of geometrical theorems

Central Angle Theorem (a.k.a. Inscribed angle property) is a special case of the Intersecting Chords Theorem.

The Intersecting Chords Theorem is a generalization of the central angle theorem.

The other Intersecting Chords Theorem says the products of the two segments of chords cut by their point of intersection are equal.

The proofs of these theorems use the Inscribed Angle property of circles:

Law of Sines - Given triangle ABC with opposite sides a, b, and c, a/(sin A) = b/(sin B) = c/(sin C) = the diameter of the circumscribed circle.

Circumscribed Circle - The radius of a circle circumscribed around a triangle is R = abc/(4K), where K is the area of the triangle.

Cyclic Quadrilateral

Ptolemy's Theorem uses the facts presented here.

The Median and Altitude of a Right Triangle are reflections about the Right Angle Bisector

Internet references

Cut-the-knot: Angles In Circle 

 

The webmaster and author of this Math Help site is Graeme McRae.