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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.
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.
