(
A
2
For an alternative formula, consider
△
I
T
C
A
{\displaystyle \triangle IT_{C}A}
r
{\displaystyle r}
r
cot
(
A
2
)
{\displaystyle r\cot \left({\frac {A}{2}}\right)}
△
I
B
′
A
{\displaystyle \triangle IB'A}
[citation needed
The Gergonne triangle (of
△
A
B
C
{\displaystyle \triangle ABC}
A
{\displaystyle A}
T
A
{\displaystyle T_{A}}
This Gergonne triangle,
△
T
A
T
B
T
C
{\displaystyle \triangle T_{A}T_{B}T_{C}}
contact triangle or intouch triangle of
△
A
B
C
{\displaystyle \triangle ABC}
K
T
=
K
2
r
2
s
a
b
c
{\displaystyle K_{T}=K{\frac {2r^{2}s}{abc}}}
where
K
{\displaystyle K}
r
{\displaystyle r}
s
{\displaystyle s}
incircle , and semiperimeter of the original triangle, and
a
{\displaystyle a}
b
{\displaystyle b}
c
{\displaystyle c}
△
T
A
T
B
T
C
{\displaystyle \triangle T_{A}T_{B}T_{C}}
contact triangle or intouch triangle of
△
A
B
C
{\displaystyle \triangle ABC}
where
K
{\displaystyle K}
r
{\displaystyle r}
s
{\displaystyle s}
incircle , and semiperimeter of the original triangle, and
a
{\displaystyle a}
b
{\displaystyle b}
c
{\displaystyle c}
extouch triangle .[21]
The three lines
A
T
A
{\displaystyle AT_{A}}
B
T
B
{\displaystyle BT_{B}The three lines
A
T
A
{\displaystyle AT_{A}}
B
T
B
{\displaystyle BT_{B}}
C
T
C
{\displaystyle CT_{C}}
Gergonne point , denoted as
G
e
{\displaystyle G_{e}}
triangle center X 7 ). The Gergonne point lies in the open orthocentroidal disk punctured at its own center, and can be any point therein.[22]
The Gergonne point of a triangle has a number of properties, including that it is the symmedian point of the Gergonne triangle.[23]
Trilinear coordinates for the vertices of the intouch triangle are given by[citation needed
Trilinear coordinates for the Gergonne point are given by[citation needed
sec
2
(
A
2
)
:
sec
2
(
B
2
)
:
sec
2
(
C
2
)
,
{\displaystyle \sec ^{2}\left({\frac {A}{2}}\right):\sec ^{2}\left({\frac {B}{2}}\right):\sec ^{2}\left({\frac {C}{2}}\right),}
or, equivalently, by the Law of Sines ,
b
c
b
+
c
−
a
:
c
a
c
+
a
−
b
:
a
b
a
+
b
−
c
.
{\displaystyle {\frac {bc}{b+c-a}}:{\frac {ca}{c+a-b}}:{\frac {ab}{a+b-c}}.}
Excircles and excenters A
triangle with
incircle,
incenter
Ior, equivalently, by the Law of Sines ,
b
c
b
+
c
−
a
:
c
a
c
+
a
−
b
:
a
b
a
+
b
−
c
.
{\displaystyle {\frac {bc}{b+c-a}}:{\frac {ca}{c+a-b}}:{\frac {ab}{a+b-c}}.}
Excircles and excenters While the incenter of
△
A
B
C
{\displaystyle \triangle ABC}
trilinear coordinates
1
:
1
:
1
{\displaystyle 1:1:1}
−
1
:
1
:
1
{\displaystyle -1:1:1}
1
:
−
1
:
1
{\displaystyle 1:-1:1}
1
:
1
:
−
1
{\displaystyle 1:1:-1}
[
A
{\displaystyle A}
external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex
A
{\displaystyle A}
excenter of
A
{\displaystyle A}
[3] orthocentric system .[5] :182
While the incenter of
△
A
B
C
{\displaystyle \triangle ABC}
trilinear coordinates
1
:
1
:
1
{\displaystyle 1:1:1}
−
1
:
1
:
1
{\displaystyle -1:1:1}
1
:
−
1
:
1
{\displaystyle 1:-1:1}
1
:
1
:
−
1
{\displaystyle 1:1:-1}
[citation needed
Exradii The radii of the excircles are called the exradii .
The exradius
The radii of the excircles are called the exradii .
The exradius of the excircle opposite
The exradius of the excircle opposite
A
{\displaystyle A}
B
C
{\displaystyle BC}
J
A
{\displaystyle J_{A}}
[25] [26]
See Heron's formula .
Derivation of exradii formula[27] Click on show to view the contents of this section
Let the excircle at side
A
B
{\displaystyle AB}
A
C
{\displaystyle AC}
G
{\displaystyle G}
Let the excircle at side
A
B
{\displaystyle AB}
A
C
{\displaystyle AC}
G
{\displaystyle G}
r
c
{\displaystyle r_{c}}
J
c
{\displaystyle J_{c}}
Then
J
c
G
{\displaystyle J_{c}G}
Then
J
c
G
{\displaystyle J_{c}G}
△
A
C
J
c
{\displaystyle \triangle ACJ_{c}}
△
A
C
J
c
{\displaystyle \triangle ACJ_{c}}
1
2
b
r
c
{\displaystyle {\tfrac {1}{2}}br_{c}}
△
B
C
J
c
{\displaystyle \triangle BCJ_{c}}
1
2
a
r
c
{\displaystyle {\tfrac {1}{2}}ar_{c}}
△
A
B
J
c
{\displaystyle \triangle ABJ_{c}}
1
2
c
r
c
{\displaystyle {\tfrac {1}{2}}cr_{c}}
Δ
{\displaystyle \Delta }
△
A
B
C
{\displaystyle \triangle ABC}
So, by symmetry, denoting
r
{\displaystyle r}
Δ
=
s
r
=
(
s
−
a
)
r
a
=
(
s
−
b
)
r
b
=
(
s
−
c
)
r
c
{\displaystyle \Delta =sr=(s-a)r_{a}=(s-b)r_{b}=(s-c)r_{c}}
cos
(
A
)
=
b
2
+
c
2
−
a
2
2
b
c
{\displaystyle \cos(A)={\frac {b^{2}+c^{2}-a^{2}}{2bc}}}
Combining this with the identity
Combining this with the identity
sin
2
A
+
cos
2
A
=
1
{\displaystyle \sin ^{2}A+\cos ^{2}A=1}
sin
(
A
)
=
−
a
4
−
b
4
−
c
4
+
2
a
2
b
2
+
2
b
But
Δ
=
1
2
b
c
sin
(
A
)
{\displaystyle \Delta ={\tfrac {1}{2}}bc\sin(A)}
Δ
=
1
4
−
a
4
−
b
4
−
c
4
+
2
a
2
b
2
+
2
b
2
c
2
+
2
a
2
which is Heron's formula .
Combining this with
s
r
=
Δ
{\displaystyle sr=\Delta }
r
2
=
Δ
2
s
2
=
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
s
.
{\displaystyle r^{2}={\frac {\Delta ^{2}}{s^{2}}}={\frac {(s-a)(s-b)(s-c)}{s}}.}
Similarly,
(
s
−
a
)
r
a
=
Δ
{\displaystyle (s-a)r_{a}=\Delta }
r
a
2
=
s
(
s
−
b
)
(
s
−
c
)
s
−
a
{\displaystyle r_{a}^{2}={\frac {s(s-b)(s-c)}{s-a}}}
and
r
a
=
s
(
s
−
b
)
(
s
−
c
Combining this with
s
r
=
Δ
{\displaystyle sr=\Delta }
Similarly,
(
s
−
a
)
r
a
=
Δ
{\displaystyle (s-a)r_{a}=\Delta }
r
a
2
=
s
(
s
−
b
)
(
s
−
c
)
s
−
a
{\displaystyle r_{a}^{2}={\frac {s(s-b)(s-c)}{s-a}}}
and
r
a
=
r
a
=
s
(
s
−
b
)
(
s
−
c
)
s
−
a
.
{\displaystyle r_{a}={\sqrt {\frac {s(s-b)(s-c)}{s-a}}}.}
[28]
Δ
=
r
r
a
r
b
r
c
.
{\displaystyle \Delta ={\sqrt {rr_{a}r_{b}r_{c}}}.}
Other excircle properties The circular hull of the excircles is internally tangent to each of the excircles and is thus an hull of the excircles is internally tangent to each of the excircles and is thus an Apollonius circle .[29]
r
2
+
s
2
4
r
{\displaystyle {\tfrac {r^{2}+s^{2}}{4r}}}
r
{\displaystyle r}
s
{\displaystyle s}
[30]
The following relations hold among the inradius
r
{\displaystyle r}
R
{\displaystyle R}
s
{\displaysThe following relations hold among the inradius
r
{\displaystyle r}
R
{\displaystyle R}
s
{\displaystyle s}
r
a
{\displaystyle r_{a}}
r
b
{\displaystyle r_{b}}
r
c
{\displaystyle r_{c}}
[13]
The circle through the centers of the three excircles has radius
2
R
{\displaystyle 2R}
[13]
If
H
{\displaystyle H}
orthocenter of
△
A
B
C
{\displaystyle \triangle ABC}
[13]
r
a
+
r
b
+
r
c
+
r
=
A
H
+
B
H
+
C
H
+
2
R
,
r
a
2
+
r
b
2
+
r
c
2
+
r
2
=
A
H
2
+
B
H
2
+
C
H
2
+
(
2
R
)
If
H
{\displaystyle H}
orthocenter of
△
A
B
C
{\displaystyle \triangle ABC}
[13]
The Nagel triangle or extouch triangle of
△
A
B
C
{\displaystyle \triangle ABC}
T
A
{\displaystyle T_{A}}
T
B
{\displaystyle T_{B}}
T
C
{\displaystyle T_{C}}
△
A
B
C
{\displaystyle \triangle ABC}
T
A
{\displaystyle T_{A}}
A
{\displaystyle A}
△
T
A
T
B
T
C
{\displaystyle \triangle T_{A}T_{B}T_{C}}
extouch triangle of
△
A
B
C
{\displaystyle \triangle ABC}
circumcircle of the extouch
△
T
A
T
B
T
C
{\displaystyle \triangle T_{A}T_{B}T_{C}}
Mandart circle .[citation needed
The three lines
A
T
A
{\displaystyle AT_{A}}
B
T
B
{\displaystyle BT_{B}}
C
T
C
{\displaystyle CT_{C}}
splitters of the triangle; they each bisect the perimeter of the triangle,[citation needed
A
B
+
B
T
A
=
A
C
+
C
T
A
=
1
2
(
A
B
+
B
C
+
A
C
)
.
{\displaystyle AB+BT_{A}=AC+CT_{A}={\frac {1}{2}}\left(AB+BC+AC\right).}
The splitters intersect in a single point, the triangle's Nagel point
N
a
{\displaystyle N_{a}}
triangle center X 8 ).
Trilinear coordinates for the vertices of the extouch triangle are given by[citation needed
Trilinear coordinates for the Nagel point are given by[citation needed
csc
2
(
A
2
)
:
csc
2
(
B
2
)
:
csc
2
(
C
2
)
,
{\displaystyle \csc ^{2}\left({\frac {A}{2}}\right):\csc ^{2}\left({\frac {B}{2}}\right):\csc ^{2}\left({\frac {C}{2}}\right),}
or, equivalently, by the Law of Sines ,
b
+
c
−
a
a
:
c
+
a
−
b
b
:
a
+
b
−
c
c
.
{\displaystyle {\frac {b+c-a}{a}}:{\frac {c+a-b}{b}}:{\frac {a+b-c}{c}}.}
The Nagel point is the isotomic conjugate of the Gergonne point.[citation needed
Related constructions Nine-point circle and Feuerbach point The nine-point circle is tangent to the incircle and excircles
In Law of Sines ,
b
+
c
−
a
a
:
c
+
a
−
b
b
:
a
+
b
−
c
c
.
{\displaystyle {\frac {b+c-a}{a}}:{\frac {c+a-b}{b}}:{\frac {a+b-c}{c}}.}
The Nagel point is the isotomic conjugate of the Gergonne point.[citation needed
isotomic conjugate of the Gergonne point.[citation needed
Related constructions Nine-point circle and Feuerbach point The nine-point circle is tangent to the incircle and excircles
In geometry , the nine-point circle is a circlIn geometry , the nine-point circle is a circle that can be constructed for any given triangle . It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are:[31] [32]
In 1822 Karl Feuerbach discovered that any triangle's nine-point circle is externally tangent to that triangle's three excircles and internally tangent to its incircle ; this result is known as Feuerbach's theorem . He proved that:[citation needed
... the circle which passes through the feet of the altitudes of a triangle is tangent to all four circles which in turn are tangent to the three sides of the triangle ... (Feuerbach 1822 )tangent to that triangle's three excircles and internally tangent to its incircle ; this result is known as Feuerbach's theorem . He proved that:[citation needed
... the circle which passes through the feet of the altitudes of a triangle is tangent to all four circles which in turn are tangent to the three sides of the triangle ... (Feuerbach 1822 ) harv error: no target: CITEREFFeuerbach1822 (help ) The triangle center at which the incircle and the nine-point circle touch is called the triangle center at which the incircle and the nine-point circle touch is called the Feuerbach point .
Incentral and excentral triangles The points of intersection of the interior angle bisectors of The points of intersection of the interior angle bisectors of
△
A
B
C
{\displaystyle \triangle ABC}
B
C
{\displaystyle BC}
C
A
{\displaystyle CA}
A
B
{\displaystyle AB}
incentral triangle . Trilinear coordinates for the vertices of the incentral triangle are given by[citation needed
<The excentral triangle of a reference triangle has vertices at the centers of the reference triangle's excircles. Its sides are on the external angle bisectors of the reference triangle (see figure at top of page ). Trilinear coordinates for the vertices of the excentral triangle are given by[citation needed
(
vertex opposite
A
)
=
−
1
:
1
:
1
{\displaystyle ({\text{vertex opposite}}\,A)=-1:1:1}
(
vertex opposite
B
)
=
1
:
−
1
:
1
{\displaystyle ({\text{vertex opposite}}\,B)=1:-1:1}
(
vertex opposite
C
)
=
1
:
1
:
−
1.
{\displaystyle ({\text{vertex opposite}}\,C)=1:1:-1.}
Equations for four circles
(
A
/
2
)
{\displaystyle u=\cos ^{2}\left(A/2\right)}
v
=
cos
2
(
B
/
2
)
{\displaystyle v=\cos ^{2}\left(B/2\right)}
w
=
cos
2
(
C
/
2
)
{\displaystyle w=\cos ^{2}\left(C/2\right)}
[33] :210–215
Incircle:
u
2
x
2
+
v
2
y
2
+
w
2
z
2
−
2
v
w
y
z
−
2
w
u
z
x
−
2
u
v
x
y
=
0
±
x
cos
(
A
2
)
±
y
cos
(
B
2
)
±
Euler's theorem states that in a triangle:
(
R
−
r
)
2
=
d
2
+
r
2
,
{\displaystyle (R-r)^{2}=d^{2}+r^{2},}
where
R
{\displaystyle R}
r
{\displaystyle r}
d
{\displaystyle d}
circumcenter and the incenter.
For excircles the equation is similar:
(
R
+
r
ex
)
2
=
d
ex
2
+
r
ex
2
,
{\displaystyle \left(R+r_{\text{ex}}\right)^{2}=d_{\text{ex}}^{2}+r_{\text{ex}}^{2},}
where
r
ex
{\displaystyle r_{\text{ex}}}
d
ex
{\displaystyle d_{\text{ex}}}
[34] [35] [36]
Generalization to other polygons Some (but not all) quadrilaterals have an incircle. These are called tangential quadrilaterals . Among their many properties perhaps the most important is that their two pairs of opposite sides have equal sums. This is called the Pitot theorem .[citation needed
More generally, a polygon with any number of sides that has an inscribed circle (that is, one that is tangent to each side) is called a tangential polygon .[citation needed
See also Notes
^ Kay (1969 , p. 140)^ Altshiller-Court (1925 , p. 74)^ a b c d e Altshiller-Court (1925 , p. 73)^ Kay (1969 , p. 117)^ a b c Johnson, Roger A., Advanced Euclidean Geometry , Dover, 2007 (orig. 1929).
^ a b Encyclopedia of Triangle Centers Archived 2012-04-19 at the Wayback Machine , accessed 2014-10-28.^ Kay (1969 , p. 201)^
Allaire, Patricia R.; Zhou, Junmin; Yao, Haishen (March 2012), "Proving a nineteenth century ellipse identity", Mathematical Gazette 96 : 161–165
^ Altshiller-Court, Nathan (1980), College Geometry , Dover Publications^ Mathematical Gazette , July 2003, 323-324.^ Chu, Thomas, The Pentagon , Spring 2005, p. 45, problem 584.
^ Kay (1969 , p. 203)^ a b c d Bell, Amy, "Hansen’s right triangle theorem, its converse and a generalization", Forum Geometricorum 6, 2006, 335–342. ^ Kodokostas, Dimitrios, "Triangle Equalizers," Mathematics Magazine 83, April 2010, pp. 141-146.
^ Allaire, Patricia R.; Zhou, Junmin; and Yao, Haishen, "Proving a nineteenth century ellipse identity", Mathematical Gazette
^ Altshiller-Court, Nathan. College Geometry , Dover Publications, 1980.
^ Posamentier, Alfred S., and Lehmann, Ingmar. The Secrets of Triangles
^ a b c Franzsen, William N. (2011). "The distance from the incenter to the Euler line" (PDF) . Forum Geometricorum . 11 : 231–236. MR 2877263 . ^ Coxeter, H.S.M. "Introduction to Geometry 2nd ed. Wiley, 1961.
^ Minda, D., and Phelps, S., "Triangles, ellipses, and cubic polynomials", American Mathematical Monthly
^
Weisstein, Eric W. "Contact Triangle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ContactTriangle.html
^ Christopher J. Bradley and Geoff C. Smith, "The locations of triangle centers", Forum Geometricorum http://forumgeom.fau.edu/FG2006volume6/FG200607index.html
^
Dekov, Deko (2009). "Computer-generated Mathematics : The Gergonne Point" (PDF) . Journal of Computer-generated Euclidean Geometry . 1 : 1–14. Archived from the original (PDF) on 2010-11-05.
^ Altshiller-Court (1925 , p. 74)^ Altshiller-Court (1925 , p. where
R
{\displaystyle R}
r
{\displaystyle r}
d
{\displaystyle d}
circumcenter and the incenter.
For excircles the equation is similar:
(
R
+
r
ex
)
2
=
d
ex
2
+
r
ex
2
For excircles the equation is similar:
where
r
ex
{\displaystyle r_{\text{ex}}}
d
ex
{\displaystyle d_{\text{ex}}}
[34] [35] [36]
Generalization to other polygons Some (but not all) quadrilaterals have an incircle. These are called tangential quadrilaterals . Among their many properties perhaps the most important is that their two pairs of opposite sides have equal sums. This is called the Pitot theorem .[citation needed
More generally, a polygon with any number of sides that has an inscribed circle (that is, one that is tangent to each side) is called a tangential polygon .[citation needed
