Theorem 4.5 The lines tangent to the circumcircle of a triangle at its vertices cut the opposite sides in three collinear points.
The proof in the text is as follows: Let the tangent to the circumcircle at A meet line BC at L. Then Angle BAL is congruent to angle C since each angle is measured by half of arc AB. *****That would be fine, but I don't know how they determine this... *****. Also we have that angle LAC = 180 - angle ABC, since these angles are measured by halves of the two opposite arcs AC. *****Again, I am lacking the theorem which is used to deduce this******.... the rest of the proof is trivial and I don't need help with it.
Can someone please give me the theorems they use for those parts of the proof.
It's elementary theorem that angles on the circumference of the circle 'looking' at the same arc of that circle are themselves equal. (the proof is a bit long but if necessary I can try to write it, try this Circumferential Angle Is Half Corresponding Central Angle ; the theorem works for any circumferential angle.)
BAL is 'looking' at the arc AB but is degenerated.
When you know this you can find the second part easily just observing the angles ABC and the angle at A 'looking' at arc AC.