Mathematical Arguments and
Triangle Geometry
Chapter 3
Coming Attractions
• Given P  Q
 Converse is Q  P
 Contrapositive is  Q   P
• Proof strategies
 Direct
 Counterexample
Deductive Reasoning
• A process
 Demonstrates that if certain statements are
true …
 Then other statements shown to follow
logically
• Statements assumed true
 The hypothesis
• Conclusion
 Arrived at by a chain of implications
Deductive Reasoning
• Statements of an argument
 Deductive sentence
• Closed statement
 can be either true or false
 The proposition
• Open statement
 contains a variable – truth value determined
once variable specified
 The predicate
Deductive Reasoning
• Statements … open? closed? true? false?
 All cars are blue.
 The car is red.
 Yesterday was Sunday.
 Rectangles have four interior angles.
 Construct the perpendicular bisector.
Deductive Reasoning
• Nonstatement – cannot take on a truth
value
 Construct an angle bisector.
• May be interrogative sentence
 Is ABC a right triangle?
• May be oxymoron
The statement in
this box is false
Universal & Existential Quantifiers
• Open statement has a variable
• Two ways to close the statement
 substitution
 quantification
• Substitution
 specify a value for the variable
x+5=9
 value specified for x makes statement either
true or false
Universal & Existential Quantifiers
• Quantification
 View the statement as a predicate or function
 Parameter of function is a value for the
variable
 Function returns True or False
Universal & Existential Quantifiers
• Quantified statement
P( x)
 All squares are rectangles x  S ,
• Quantifier = All
• Universe = squares
• Must show every element of universe has the
property of being a square
 Some rectangles are not squares
• Quantifier = “there exists” x  R,
• Universe = rectangles
P ( x )
Universal & Existential Quantifiers
• Venn diagrams useful
in quantified statements
• Consider the definition
of a trapezoid
 A quadrilateral with a pair of parallel sides
 Could a parallelogram be a trapezoid
according to this diagram?
• Write quantified statements based on this
diagram
Negating a Quantified Statement
• Useful in proofs
P  Q  Q  P
 Prove a statement false   P  Q   P  Q
 Prove the contrapositive
• Negation patterns for quantified statements
 x, P( x)   x, P( x)
 x, P( x)   x, P( x)
Try It Out
• Negate these statements
 Every rectangle is a square
 Triangle XYZ is isosceles, or a pentagon is a
five-sided plane figure
 For every shape A, there is a circle D such
that D surrounds A
 Playfair’s Postulate:
Given any line l, there is exactly one line m
through P that is parallel to l (see page 41)
Proof and Disproof
• Start by being clear about assumptions
 Euclid’s postulates are implicit
• Clearly state conjecture/theorem
 What are givens, the hypothesis
 What is conclusion
P Q
Proof and Disproof
• Direct proof
 Work logically forward
 Step by step
 Reach logical (and desired) conclusion
• Use Syllogism
 If P  Q and Q  R and R  S are
statements in a proof
 Then we can conclude P  S
Proof and Disproof
• Counterexample in a proof
 All hypotheses hold
 But discover an example where conclusion
does not
• This demonstrates the conjecture to be
false
• Counterexample suggests
 Alter the hypotheses … or …
 Change the conclusion
Step-By-Step Proofs
• Each line of proof
 Presents new idea, concept
 Together with previous steps produces new
result
• Text suggests
 Write each line of proof as complete sentence
 Clearly justify the step
• Geogebra diagrams are visual
demonstrations
Congruence Criteria for Triangles
• SAS: If two sides and the included angle
of one triangle are congruent to two sides
and the included angle of another triangle,
then the two triangles are congruent.
• We will accept this axiom without proof
Angle-Side-Angle Congruence
• State the Angle-Side-Angle criterion for
triangle congruence (don’t look in the
book)
• ASA: If two angles and the included side
of one triangle are congruent respectively
to two angles and the included angle of
another triangle, then the two triangles are
congruent
Angle-Side-Angle Congruence
• Proof
• Use negation
• Justify the steps in the proof on next slide
ASA
• Assume
AB DE
x  DE AB  DX
ABC  DXF
C  XFD
But given C  EFD
AB  DX  DE
ABC  DEF
Incenter
• Consider the angle bisectors
• Recall Activity 6
• Theorem 3.4
The angle bisectors of a triangle are
concurrent
Incenter
Proof
• Consider angle bisectors for angles A and
B with intersection point I
• Construct
perpendiculars
to W, X, Y
• What congruent
triangles do you see?
• How are the perpendiculars related?
Incenter
• Now draw CI
• Why must it bisect angle C?
• Thus point I is concurrent to all three angle
bisectors
Incenter
• Point of concurrency called “incenter”
 Length of all three perpendiculars is equal
 Circle center at I, radius equal to
perpendicular is incircle
Viviani’s Theorem
• IF a point P is interior to
an equilateral triangle THEN the sum of the
lengths of the perpendiculars from P to the sides
of the triangle is equal to the altitude.
Viviani’s Theorem
• What would make the hypothesis false?
• With false hypothesis, it still might be
possible for the lengths to equal the
altitude
Converse of Viviani’s Theorem
• IF the sum of the lengths of the
perpendiculars from P to the sides of the
triangle is equal to the altitude
THEN a point P is interior to
an equilateral triangle
• Create a counterexample to this converse
Contrapositive
• Recall Given P  Q
 Contrapositive is  Q   P
• These two statements are equivalent
 They mean the same thing
 They have the same truth tables
• Contrapositive a valuable tool
 Use for creating indirect proofs
Orthocenter
• Recall Activity 4
• Theorem 3.8 The altitudes of a triangle
are concurrent
Centroid
• A median : the line segment from the
vertex to the midpoint of the opposite side
• Recall Activites
Centroid
• Theorem 3.9 The three medians of a
triangle are concurrent
• Proof
 Given ABC, medians AD
and BE intersect at G
 Now consider midpoint
of AB, point F
Centroid
• Draw lines EX and
FY parallel to AD
• List the pairs of
similar triangles
• List congruent
segments on side CB
• Why is G two-thirds of the way along
median BE?
Centroid
• Now draw median
CF, intersecting
BE at G’
• Draw parallels as
before
• Note similar triangles and the fact that G’
is two-thirds the way along BE
• Thus G’ = G and all three medians
concurrent
Circumcenter
• Recall Activities
• Theorem 3.10
The three perpendicular bisectors of the
sides of a triangle are concurrent.
 Point of concurrency called circumcenter
• Proof left as an exercise!
Ceva’s Theorem
• A Cevian is a line segment from
the vertex of a triangle to a point
on the opposite side
 Name examples of Cevians
• Ceva’s theorem for triangle ABC
 Given Cevians AX, BY, and CZ concurrent
 Then
AZ BX CY


1
ZB XC YA
Ceva’s Theorem
Proof
• Name similar
triangles
• Specify resulting
ratios
• Now manipulate algebraically to arrive at
product equal to 1
Converse of Ceva’s Theorem
• State the converse of the theorem
 If
AZ BX CY


1
ZB XC YA
 Then the Cevians are concurrent
• Proving uses the contrapositive of the
converse
 If the Cevians are not concurrent
 Then
AZ BX CY


1
ZB XC YA
Preview of Coming Attractions
Circle Geometry
• How many points to determine a circle?
• Given two points … how many circles can
be drawn through those two points
Preview of Coming Attractions
• Given 3 noncolinear points … how many
distinct circles can be drawn through these
points?
 How is the construction done?
 This circle is the circumcircle of triangle ABC
Preview of Coming Attractions
• What about four points?
 What does it take to guarantee a circle that
contains all four points?
Nine-Point Circle (First Look)
• Recall the orthocenter,
where altitudes meet
• Note feet of the altitudes
 Vertices for the pedal
triangle
• Circumcircle of pedal triangle
 Passes through feet of altitudes
 Passes through midpoints of sides of ABC
 Also some other interesting points … try it
Nine-Point Circle (First Look)
• Identify the different lines and points
• Check lengths of diameters
Mathematical Arguments and
Triangle Geometry
Chapter 3