What is the Unknown Theorem Used in the Proof of Theorem 4.5?

[Noxide]

I will preface this by saying that if anyone has the following book: Euclidean Geometry and Transformations written by Clayton W. Dodge, then my question concerns a theorem used but unstated in the proof of Theorem 4.5.

Theorem 4.5
The lines tangent to the circumcircle of a triangle at its vertices cut the opposite sides in three collinear points.

The portion of the proof for which I am having a little trouble is the following:
Let the tangent to the circumcircle at A (triangle vertices are named counterclockwise) meet line BC at L. Then angle BAL is congruent to angle C since each angle is measured by half of the arc AB. Also we have that angle LAC = 180 - angle ABC, since these angles are measured by halves of the two opposite arcs AC...

Specifically my trouble lies with: angle LAC = 180 - angle ABC, since these angles are measured by halves of the two opposite arcs AC.

The theorem from which this was determined is unknown to me, and I would greatly appreciate any help in locating it, or having it stated.

 

[tiny-tim]

Hi Noxide!

It's using the theorem that if A B and C are on a circle with centre O, then angle ACB is either half of angle AOB, or is 180° minus that.

 

[Noxide]

Hmm. I don't know if that one applies here. My geometry is really weak and I would appreciate any possible insight.

I'll add the image below. Notice that L is a proper Cevian.

http://img13.imageshack.us/img13/9553/tricircle.png

Uploaded with ImageShack.us

 

[tiny-tim]

(just got up :zzz: …)

Hint: draw the diameter AOD.

 

[Noxide]

Hmm. I don't think I know how to get that angle BAL is congruent to angle C afterall...

 

[tiny-tim]

(just got up :zzz: …)

Have you drawn diameter AOD?

What is angle ADC equal to? ​