1. 6.4 Some Constructions and Inequalities for the Circle.
Constructing a tangent to a circle
Theorem 6.4.1: The line that is perpendicular to the radius of a circle at its endpoint on the circle is a tangent to the circle.
Strategy: Draw a radius and extend it beyond the circle. At the point on the circle where the radius touches the circle, construct a line perpendicular to the radius.
2/23/2019
Section 6.4 Nack

2. Constructing a Tangent to a Circle
Draw radius PX and
extend it to form PX.
2. Using X as the center,
construct a line perpendicular to PX. 
3. That line, XW is the tangent to Circle P. X P W ( X ) P
2/23/2019
Section 6.4 Nack

3. Construct a tangent to a circle from an external point
Given Q and external point E
Construct: Tangent ET with T as the point of tangency.
Draw EQ.
Construct the bisector EQ to intersect EQ at its midpoint M.
With ME as the length of the radius, construct a circle.
Draw ET, the desired tangent.
Note: EV would also be a tangent.
E M Q  T
E M Q  V
2/23/2019
Section 6.4 Nack

4. Inequalities in the Circle
Theorem 6.4.2: In a circle ( or in congruent circles) containing two unequal central angles, the larger angle corresponds to the larger intercepted arc. Proof p. 307
Theorem 6.4.3: In a circle ( or in congruent circles) containing two unequal arcs, the larger arc corresponds to the larger central angle.
Theorem 6.4.4: In a circle ( or in congruent circles) containing two unequal chords, the shorter chord is at a greater distance from the center of the circle. Fig. 6.59
Theorem : In a circle ( or in congruent circles) containing two unequal chords, the chord nearer the center of the circle has the greater length. Fig. 6.60
2/23/2019
Section 6.4 Nack

5. Theorems on Minor Arcs
Theorem 6.4.6: In a circle ( or in congruent circles) containing two unequal chords, the longer chord corresponds to the greater minor arc.
Theorem 6.4.7: In a circle ( or in congruent circles) containing two unequal chords, the greater minor arc corresponds to the longer of the chords related to these arcs. Fig (four cases)
2/23/2019
Section 6.4 Nack