Summary of the Concurrence Theorems

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10 Οκτ 2013 (πριν από 3 χρόνια και 11 μήνες)

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H
-
Geometry

Jacobs’ Chapter 13 / The Concurrence Theorems

Summary of the Concurrence Theorems


Definition
: A
cevian

of a triangle is a line segment that joins a vertex of the triangle to a point on
the opposite side.


This leads to Ceva’s Theorem.


Theorem
: Three cevians of a triangle are concurrent

if and only

if

or



(in the diagram)



We have seen three types of cevians:
altitudes
,
angle bisectors

and
medians
.


Properties of Altitudes
: 1) The lines containing the altitudes of a triangle are concurrent.

2) The point of concurrency for the lines containing the altitudes is called
the
orthocenter

of the triangle.

Note
: The orthocenter of a triangle is not always inside the triangle. You
can see this by drawing an acute, right and obtuse triangles and
then
sketching their altitudes.


Properties of Angle Bisectors:

1) The angle bisectors of a triangle are concurrent.

2) The point of concurrency of the angle bisectors is called the
incenter

of the triangle.

3) The incenter is the center of the
incircle
of th
e triangle (the
incircle is the inscribed circle for the triangle).

4) As a result, the incenter is the point that is equidistant from all
3 sides of the triangle.


Properties of Medians
: 1) The medians of a triangle are concurrent.

2) The point of concur
rency for the medians is called the
centroid

of the
triangle.

3) The centroid of a triangle divides a median into two segments who
se
lengths are in the ratio 2:1 (
not in book
).



The perpendicular bisectors of the sides of a triangle are NOT cevians. Howe
ver, the
perpendicular bisectors of the sides of a triangle do have interesting properties.


Properties of Perpendicular Bisectors
: 1) The perpendicular bisectors of the sides of a triangle are
concurrent.

2) The point of concurrency of the perpendicular

bisectors
is called the
circumcenter

of the triangle.

3) The circumcenter is the center of the
circumcircle

(the
circle circumscribed about the triangle).

4) As a result, the circumcenter is the point that is
equidistant from all 3 vertices of the triangl
e.

a


e

d

c

b