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

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SUMMARY

The discussion centers on the proof of Theorem 4.5 from Clayton W. Dodge's book "Euclidean Geometry and Transformations," which states that the lines tangent to the circumcircle of a triangle at its vertices intersect the opposite sides at three collinear points. The user seeks clarification on the relationship between angles LAC and ABC, specifically that angle LAC equals 180° minus angle ABC, which is derived from a theorem about angles subtended by arcs in a circle. The theorem in question is that if points A, B, and C lie on a circle with center O, then angle ACB is either half of angle AOB or 180° minus that angle.

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Noxide
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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.
 
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Hi Noxide! :wink:

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. :smile:
 
Last edited by a moderator:
(just got up :zzz: …)

Hint: draw the diameter AOD. :wink:
 
Hmm. I don't think I know how to get that angle BAL is congruent to angle C afterall...
 
(just got up :zzz: …)

Have you drawn diameter AOD?

What is angle ADC equal to? :smile:​
 

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