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2
The Three Kinds of Circles Associated with a Triangle (thing)
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(
thing
)
by
IWhoSawTheFace
Sat Nov 26 2011 at 19:15:34
Reference Triangle
ΔABC
Vertices
A
,
B
, and
C
Side lengths a, b, c
Interior angles α, β, γ
Semiperimeter
s = (a+b+c)/2
Area K = sqrt(s⋅(s-a)⋅(s-b)⋅(s-c))
Inscribed Circle
Radius
: r = K/s
Center
: 1:1:1 (trilinear)
O
= (Ox,Oy) = ( (aAx+bBx+cCx)/(a+b+c), (aAy+bBy+cCy)/(a+b+c) )
The
incenter
is the intersection of the three lines of angle bisection
The
incircle
touches the triangle at three points
From vertices
A
,
B
, and
C
, the distances to the points of tangency are s-a, s-b, and s-c, respectively
Circumscribed Circle
Radius
: R = (a⋅b⋅c/(4⋅K)
Center
: cos(α):cos(β):cos(γ) (trilinear)
O
= (Ox,Oy) = ( see
circumcenter
)
The
circumcenter
is the intersection of the three lines of side bisection, i.e., perpendicular bisectors of the sides
The
circumcircle
touches the triangle at three points
These points are the vertex points
A
,
B
, and
C
From vertices
A
,
B
, and
C
the distances to the points of tangency (along the lines AB, BC, and CA) are c, a, and b, respectively
Exscribed Circles
There are three
excircles
for every triangle, denoted Γ
A
, Γ
B
, and Γ
C
Radius
: rA = K/(s-a), rB = K/(s-b), rC = K/(s-c)
Center
:
O
A
: -1:0:0,
O
B
: 0:-1:0,
O
C
: 0:0:-1, (trilinear)
Each
excenter
is the intersection of three angle bisectors. Excenter
O
A
, for example, lies at the intersection of the interior angle bisector of
A
plus the two exterior angle bisectors of
B
, and
C
The
circumcircle
touches the reference triangle at one point and the extensions of the other two sides at two other points.
The distances to the points of tangency are:
For
O
A
:
From vertex
A
, along the line AB: s
From vertex
B
, along the line CB: s-c
From vertex
C
, along the line AB: s-b
For
O
B
:
From vertex
B
, along the line BC: s
From vertex
C
, along the line CA: s-a
From vertex
A
, along the line AB: s-c
For
O
C
:
From vertex
C
, along the line CA: s
From vertex
A
, along the line AB: s-b
From vertex
B
, along the line BC: s-a
K = sqrt(r⋅rB⋅rC⋅rA)
K is the area of the reference triangle
The areas of the extriangles are:
K
A
= (TBD)
K
B
= (TBD)
K
C
= (TBD)
1/r = 1/rA + 1/rB + 1/rC
ScienceQuest 2012
THE IRON NODER CHALLENGE 4: FERRASSIC PARK
Semiperimeter
circumcircle
Crockett Johnson
Incircle
Incenter
Circumcenter
vertex