Geometry Proof (Circles and Chords)

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    Geometry Proof
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SUMMARY

The discussion focuses on proving the theorem that states the angle formed by a tangent and a chord through the point of contact is equal to half the measure of the intercepted arc. Specifically, the proof involves demonstrating that angle ABC equals half of arc BC, utilizing the properties of tangents and triangles. Key steps include drawing lines from the center of the circle to the endpoints of the chord and applying the angle sum formula for triangles. The discussion emphasizes the importance of visual representation in geometric proofs.

PREREQUISITES
  • Understanding of basic circle theorems, particularly tangent and chord properties.
  • Familiarity with triangle angle sum properties.
  • Ability to construct geometric figures accurately.
  • Knowledge of Euclidean geometry principles.
NEXT STEPS
  • Study the properties of tangents and chords in circles.
  • Learn how to apply the angle sum formula in triangle proofs.
  • Explore Euclid's Elements, particularly Book III on circle theorems.
  • Practice constructing geometric proofs with various configurations of circles and angles.
USEFUL FOR

Students studying geometry, particularly those preparing for tests involving circle theorems and geometric proofs, as well as educators seeking to enhance their teaching methods in geometry.

Volux
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Hey all. At my school we have these things called "Real Problems" that we get every 3-4 weeks. This time it's about proofs. I have an A- in the class, so I got the difficult one with 4 very difficult (IMO) proofs.

I have NO idea where to start on this..I have drawn it out for you. All help is greatly appreciated. I am here to merely learn, not copy everything. I would appreciate full answers, but I am definitely here to Learn how to do this for tests.

Given: Tangent line AB and chord BC.
Prove: <ABC = (1/2)(arc)BC

In other words prove: (Angle ABC = Half of arc BC)

HINT: Using the center of the circle, draw in OB and OC (which I did in the picture for you).

[PLAIN]http://img249.imageshack.us/img249/4413/proof1f.jpg
 
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you might look in euclid where this is a standard theorem, probably in book III.

by the way you have probably started it more or less correctly, as the first step in a circle proof is always to draw an appropriate line to the center.

in fact the proof is immediate from your figure if you know the angle sum formula for a triangle, and the fact that a tangent is perpendicular to the radius.
 
Consider drawing a line segment from the center of the circle, through the midpoint of BC, until it intersects AB.
 

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