Angles In Circles
Made From Tangents, Secants, Radii, Chords


An inscribed angle consists of two intersecting chords which share an endpoint on the circle.
A central angle consists of two radii with a vertex at the circle's center.



When a tangent line (BC) intersects with a chord (BA), it forms an angle (∠d) equal to one half of the intercepted arc.


When two tangent lines intersect, they form an angle (∠y) equal to:
      ½ (the larger intercepted arc minus the smaller intercepted arc)

The two tangent lines (xy & zy) will always be of equal length.


When a tangent (Line C) and a secant (Lines B + A) intersect, they form an angle (∠y) equal to:
      ½ (the larger intercepted arc minus the smaller intercepted arc)

The tangent line squared (C2) equals the external secant line segment (A) times the length of the entire secant (A + B).

C2 = A • (A + B)



When two secants (Line A + B and Line C + D) intersect, they form an angle (∠y) equal to:
      ½ (the larger intercepted arc minus the smaller intercepted arc)


The line segments they form can be calculated from this formula:

(A + B) • B = (C + D) • D



Whenever two chords of a circle intersect, they form two pairs of vertical angles:
1) Angles e & f and 2) Angles g & h.
Angles e and f, will each equal half the sum of the intercepted arcs:
∠e = ∠f = ½ (arc AC + arc BD)
Using the example in the graphic: ∠e = ∠f = ½ (31° + 69°) = 50°

We know there are 360° in a circle so (∠g + ∠h) = 360° minus (∠e + ∠f)
(∠g + ∠h) = 360° minus (50° + 50°)   (∠g + ∠h) = 260°
∠g = 130° ∠h =130°


Whenever two chords intersect, the products of their segments are equal.
Looking at the graphic, we see 2 chords intersecting at point "E".

When two chords intersect,
      chord segments (AE • BE) = (CE • DE)

As we can easily see, 8 • 3 = 6 • 4



An inscribed quadrilateral (or a cyclic quadrilateral) has all four of its vertices resting on a single circle.
Opposite angles in an inscribed quadrilateral are supplementary (sum to 180°)
∠A + ∠C = 180°
∠B + ∠D = 180°


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