Two Secants
A secant of a circle is a line connecting two points on the circle. When two nonparallel secants are drawn, a number of useful properties are satisfied, even if the two intersect outside the circle.
These properties are especially useful in the context of cyclic quadrilaterals, as they often allow various angles and/or lengths to be filled in. In fact, these results are so useful that it is not unusual to add lines to a diagram for the purpose of creating two-secant configurations.
Intersection Inside the Circle
When the two secants intersect inside the circle, they can be viewed as diagonals of a cyclic quadrilateral, and as such they satisfy the same properties. Let and be the two secants, intersecting inside the circle at point . The first major property is that pairs of angles subtending the same arc are equal, as per the inscribed angle theorem, which gives four important angle equalities:
, since they both subtend arc . Similarly, , , and , by looking in turn at the arcs and .
A similar property is that opposite angles add to , again due to the inscribed angle properties.
The final important property is the first case of the power of a point theorem, which states
Intersection Outside the Circle
When the two secants intersect outside the circle, they can be viewed as the extensions of two sides of a cyclic quadrilateral, and as such they satisfy the same properties. Let and be the two secants, intersecting inside the circle at point . The first major property is that pairs of angles subtending the same arc are equal, as per the inscribed angle theorem, which gives four important angle equalities:
, since they both subtend arc . Similarly, , , and , by looking in turn at the arcs and .
Again, opposite angles add to :
The final important property is the second case of the power of a point theorem, which states
Note that this is the same statement as in the previous section. In fact, the power of a point with respect to a circle depends only on its distance to the origin; it doesn't matter whether the point is inside or outside the circle. Specifically, the power of point is , where is the radius of circle , and this is independent of the secant chosen.