Geometry 5 Flashcards
circumcenter
intersection of the perpendiular bisectors of the sides of a triangle
incenter
intersection of the angle bisectors of the vertices of the triangle
concurrent lines?
3 or more lines intersecting at one point
Is the incenter ever outside of the triangle?
No
When can the circumcenter be outside the triangle?
if the triangle is obtuse
When can the circumcenter of the triangle be on a side
if the triangle is right
When can the circumcent be inside the triangle?
when it is acute
What is the circumcenter always equidistant to?
the vertices
What is the incenter always equidistant to?
the sides
when is the incenter equidistant to the vertices/ or the circumcenter equidistant to the sides
if the triangle is equilateral.
incircle versus circumcircle
the circle made from the intersections of the lines perpendicular to the sides is the incircle-
The circle made from the vertices of a triangle is the circumcircle.
5.1&5.2 Perpendicular bisector theorem&converse
If a point is on the perpendicular bisector of a segment then its equidistant from the endpoints.
5.5&5.4 Angle bisector theorem&converse.
If a point is on the angle bisector of an angle, it is equidistant from the angle sides
Circumcenter theorem
The circumcenter of a triangle is equidistant from the vertices
Incenter theorem
The incenter of a triangle is equidistant from it’s sides.
medians
lines in a triangle from a vertax to the mid point of the opposite side.
altitudes
A line showing the distance of a vertex from the opposite line, altitude is perpendicular to opposite line.
Centroid
Intersection of the medians! ALWAYS inside triangle.
-2/3s the distance from the vertex t the opposite side (2/3s the way on the median)
Orthocenter
intersection of all the altitudes.
What do
-Altitudes
-Medians
- perpendicular bisectors
- angle bisectors
intersect to form?
- altitudes- orthocenter
- medians- centroid (center of gravity)
- perpendicular bisectors- circumcenter
- angle bisector- incenter
What is the definition of inequalities
5=2+3
THEN
5 is greater than 2 and 3
Theorem 5.8(Exterior angle inequality)
because m^1+m^2=m^3
THEN
m^1<m^3
and
m^2<m^3
Theorem 5.9&5.10
-ASA relationships
- AB>BC so m^B> m^A
- CONVERSE
comparison (properties of real numbers)
a<b, a=b, OR a>b
5.11 Triangle inequality theorem
sides a,b and c
a+b<c
b+c<a
c+a<b
what is an indirect proof
