Test 2 Flashcards
A statement with a variable, also known as a open statement
predicate
statement with a variable
open statement
a statement either true or false; sometimes called a proposition
closed statement
a way to close an open statement, choosing variable conditions
quantification
existential quantifier; “exists” “some” or “at least one”
∃
“in”
∈
a universal quantifier; “all” or “every”
∀
“not” “all x’s” are P(x) = there is at least one x not P(x)
~(∀x)P(x)≡(∃x)~P(x)
~(∃x)P(x)≡(∀x)~P(x)
there is not at least one x that are P(x) = all x are no P(x)
Chain of implications leading directly from hypothesis to conclusion
direct proof
rule of logic that moves a proof forward ina directly way
syllogism
disproof of a conjecture
counterexample
(points) lying in the same straight line
collinearity
the line segment from a vertex perpendicular to the line containing the opposite side
altitude
the point at which the three altitudes of a triangle intersect (H)
orthocenter
line segment from a vertex to the midpoint of the opposite side
median
the point where 3 medians of a triangle intersect (G)
centroid
segment from vertex of a triangle to a point on the line containing the opposite side
cevian
center at the circumcenter and passes through the vertices of the triangle
circumcircle
the point where all 3 perpendicular bisectors of a triangle intersect
circumcenter
the point at which the angle bisectors of a triangle intersect
incenter
the set of points at a fixed distance, r, from a fixed point, O
circle
the fixed point, O, in a circle
center
the distance, r, in a circle
radius
line segment joining two points on a circle
chord
line that intersects at exactly one point
tangent
the point where a tangent line touched the circle
point of tangency
the perimeter of a circle
circumference
a piece of the circle
arc
a pie-shaped portion of the interior of the circle, bounded by an arc of the circle and two radii
sectors
if P, Q, and R are three points on a circle with the center at O, the angle <PQR
central angle
if P, Q, and R are three points on a circle with center at O, the angle <PQR
inscribed angle
the center of the incircle, I
incenter
a circle interior to the triangle that is tangent to all three sides of the triangle
incircle
a circler exterior to the triangle and tangent to one side and to extensions of the other two sides
excircle
If two circles’ tangents are perpendicular at their points of intersection
orthogonal
the region bounded by the three semicircular arcs on one side of the diameter AB of a circle
arbelos
another “father” of geometry, authored Foundations of Geometry, also known for defining Hilbert’s space and organizing Euclid’s axioms into five groups
David Hilbert
specific exactly what is meant by a point is “on a line,” a line “goes through a point,” or a line “lies in a plane”
axiom of incedence
how we know when a point is between two other points, or a ray is between two other rays
axiom of betweeness
tools for developing proofs, used to apply theorems to particular situations
modus ponens
tool for developing proofs, prove the contrapositive of the conjecture (indirect proof)
modus tollens
preliminary result needed to prove a particular theorem
lemma
a result follows fairly easily from a previous result
corollary
power of P with respect to C, d^2-r^2
Power (P,C)
line formed from the set of points P for which the power is the same value from both circles
Radical Axis
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.
axiom of congruence
