圆周角定理：切线的情况

Intuitive limiting argument

Sometimes we have a sort of "degenerate/tangent" inscribed angle, like $\angle{ABX}$ below. It's a "degenerate tangent case" in the sense that the inscribed angle is formed by the secant $AB$ and the tangent $BX$ to the magenta circle (when the circle in question is clear, it can be helpful, at least intuitively, to call the tangent line "$BB$", and the inscribed angle "$\angle{ABB}$").

Do you have a guess for what $\angle{ABX}$ might equal, in view of the blue, green, yellow, and orange lines (and the previous topic on "normal" inscribed-angle/intercepted-arc relationships)?

In the diagram, it looks like the sequence of inscribed angles $$\angle{ABP}, \angle{ABQ}, \angle{ABR}, \angle{ABS}$$ "approaches the limiting degenerate/tangent inscribed angle $\angle{ABB} = \angle{ABX}$".

Similarly, it looks like the sequence of arcs $$\widehat{AP}, \widehat{AQ}, \widehat{AR}, \widehat{AS}$$ "approaches the arc $\widehat{AB}$" (although this time we don't have any "degeneracy" or anything like that).

Based on the previously encountered "normal case", we have $$\begin{align*} \angle{ABP} &= \frac12 \widehat{AP}, \\ \angle{ABQ} &= \frac12 \widehat{AQ}, \\ \angle{ABR} &= \frac12 \widehat{AR}, \\\ \angle{ABS} &= \frac12 \widehat{AS} \end{align*}$$ so might expect that $\angle{ABX} = \angle{ABB}$ is simply equal to $\frac12 \widehat{AB}$.

This is indeed true, but in order to "formalize" the argument above, we need to use a basic tool/notion from calculus---that of limits and continuity. But don't worry about this---if you can understand the intuition, that's great.

Caveat with "lazy $\angle{ABB}$" notation

If we "approach $B$ from the other direction", then we need to be careful talking about "$\angle{ABB}$". In this new context we would mean $\angle{ABB} = \angle{ABY}$, which is supplementary, so generally unequal, to $\angle{ABX}$.

There are some useful formulas to calculate angles in circles, formed by radii, chords, secants, and tangents.

1. Central Angle ========

A central angle is formed by two intersecting radii in a circle. Its vertex must be at the center of the circle.

In this diagram, x is the central angle. In a circle, or congruent circles, congruent central angles have congruent arcs.

Central Angle = Intercepted Arc

Thus,

Angle Measure of Arc AB = Central Angle x (Angle AOB)

1. Inscribed Angle =====

An inscribed angle is an angle with its vertex on the circumference of the circle. It is formed by two intersecting chords.

Inscribed Angle = 1/2 Intercepted Arc

Thus,

1/2*Angle Measure of Arc AC = Angle ABC

1. Tangent Chord Angle =====

An angle formed by an intersecting tangent and chord; the angle has its vertex on the circumference of the circle.

Tangent Chord Angle = 1/2 Intercepted Arc

Angle ABC= 1/2 Measure of Arc AB

1. Angle Formed Inside of a Circle By Two Intersecting Chords ======

When two chords intersect inside a circle, they created two sets of vertical angles (vertical angles are always equal). In the diagram, angle x is on of the vertical angles.

Angle Formed Inside Circle by Intersecting Chords = 1/2 Sum of Intercepted Arcs

Measure of Angle BED = 1/2 (Angle Measure of Arc AC + Angle Measure of Arc BD)

1. Angle Formed By the Intersection of Two Tangents, Two Secants, or a Tangent and a Secant: ========

The formula for ALL THREE of these cases are the same:

Angle Formed Outside Circle = |1/2 Difference of Intercepted Arcs|

a) Two Tangents

Angle ABC is formed by two tangents intersecting outside of the circle at point B. The intercepted arcs are minor arc AC and major arc AC.

b) Two Secants

Angle ACE is formed by two secants intersecting at point C. The intercepted arcs in this diagram are the minor arcs BD and AE.

c) A Secant and a Tangent

Angle ABD is formed by the tangent and secant intersecting at point B. The intercepted arcs in the diagram are the minor arcs AC and AD.

References: "Formulas for Working with Angles in Circles." Geometry Lesson Page. Regentsprep.org, 1988. Web. 6 Sept. 2014.

Use the "tangent case formula" to derive the "normal case formula".

(This is not circular, because it's possible to prove the tangent case formula without using the normal case formula---in fact, it's easier that way.)

Here is a worksheet for identifying tangents: http://cdn.kutasoftware.com/Worksheets/Geo/11-Tangents to Circles.pdf

Problems 1-6 pertain specifically to this topic, but the others are fun as well

The Pythagorean Theorem, $c^2 = a^2 + b^2$, is useful for these exercises.