(
A
2
For an alternative formula, consider
△
I
T
C
A
{\displaystyle \triangle IT_{C}A}
. This is a right-angled triangle with one side equal to
r
{\displaystyle r}
and the other side equal to
r
cot
(
A
2
)
{\displaystyle r\cot \left({\frac {A}{2}}\right)}
. The same is true for
△
I
B
′
A
{\displaystyle \triangle IB'A}
. The large triangle is composed of six such triangles and the total area is:[citation needed ]
The Gergonne triangle (of
△
A
B
C
{\displaystyle \triangle ABC}
) is defined by the three touchpoints of the incircle on the three sides. The touchpoint opposite
A
{\displaystyle A}
is denoted
T
A
{\displaystyle T_{A}}
, etc.
This Gergonne triangle,
△
T
A
T
B
T
C
{\displaystyle \triangle T_{A}T_{B}T_{C}}
, is also known as the contact triangle or intouch triangle of
△
A
B
C
{\displaystyle \triangle ABC}
. Its area is
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}
, and
s
{\displaystyle s}
are the area, radius of the incircle , and semiperimeter of the original triangle, and
a
{\displaystyle a}
,
b
{\displaystyle b}
, and
c
{\displaystyle c}
This Gergonne triangle,
△
T
A
T
B
T
C
{\displaystyle \triangle T_{A}T_{B}T_{C}}
, is also known as the contact triangle or intouch triangle of
△
A
B
C
{\displaystyle \triangle ABC}
. Its area is
where
K
{\displaystyle K}
,
r
{\displaystyle r}
, and
s
{\displaystyle s}
are the area, radius of the incircle , and semiperimeter of the original triangle, and
a
{\displaystyle a}
,
b
{\displaystyle b}
, and
c
{\displaystyle c}
are the side lengths of the original triangle. This is the same area as that of the 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}}
and
C
T
C
{\displaystyle CT_{C}}
intersect in a single point called the Gergonne point , denoted as
G
e
{\displaystyle G_{e}}
(or 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
I 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
While the incenter of
△
A
B
C
{\displaystyle \triangle ABC}
has trilinear coordinates
1
:
1
:
1
{\displaystyle 1:1:1}
, the excenters have trilinears
−
1
:
1
:
1
{\displaystyle -1:1:1}
,
1
:
−
1
:
1
{\displaystyle 1:-1:1}
, and
1
:
1
:
−
1
{\displaystyle 1:1:-1}
.[
A
{\displaystyle A}
, for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex
A
{\displaystyle A}
, or the excenter of
A
{\displaystyle A}
.[3] Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle together with the three excircle centers form an orthocentric system .[5] :182
While the incenter of
△
A
B
C
{\displaystyle \triangle ABC}
has trilinear coordinates
1
:
1
:
1
{\displaystyle 1:1:1}
, the excenters have trilinears
−
1
:
1
:
1
{\displaystyle -1:1:1}
,
1
:
−
1
:
1
{\displaystyle 1:-1:1}
, and
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}
(so touching
B
C
{\displaystyle BC}
, centered at
J
A
{\displaystyle J_{A}}
) is[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}
touch at side
A
C
{\displaystyle AC}
extended at
G
{\displaystyle G}
, and let this excircle's
radius be Let the excircle at side
A
B
{\displaystyle AB}
touch at side
A
C
{\displaystyle AC}
extended at
G
{\displaystyle G}
, and let this excircle's
radius be
r
c
{\displaystyle r_{c}}
and its center be
J
c
{\displaystyle J_{c}}
.
Then
J
c
G
{\displaystyle J_{c}G}
is an altitude of
Then
J
c
G
{\displaystyle J_{c}G}
is an altitude of
△
A
C
J
c
{\displaystyle \triangle ACJ_{c}}
,
so
△
A
C
J
c
{\displaystyle \triangle ACJ_{c}}
has area
1
2
b
r
c
{\displaystyle {\tfrac {1}{2}}br_{c}}
.
By a similar argument,
△
B
C
J
c
{\displaystyle \triangle BCJ_{c}}
has area
1
2
a
r
c
{\displaystyle {\tfrac {1}{2}}ar_{c}}
and
△
A
B
J
c
{\displaystyle \triangle ABJ_{c}}
has area
1
2
c
r
c
{\displaystyle {\tfrac {1}{2}}cr_{c}}
.
Thus the area
Δ
{\displaystyle \Delta }
of triangle
△
A
B
C
{\displaystyle \triangle ABC}
is
So, by symmetry, denoting
r
{\displaystyle r}
as the radius of the incircle,
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}}
Law of Cosines, we have
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}
, we have
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)}
, and so
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 }
, we have
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 }
gives
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 }
, we have
Similarly,
(
s
−
a
)
r
a
=
Δ
{\displaystyle (s-a)r_{a}=\Delta }
gives
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] The radius of this Apollonius circle is
r
2
+
s
2
4
r
{\displaystyle {\tfrac {r^{2}+s^{2}}{4r}}}
where
r
{\displaystyle r}
is the incircle radius and
s
{\displaystyle s}
is the semiperimeter of the triangle.[30]
The following relations hold among the inradius
r
{\displaystyle r}
, the circumradius
R
{\displaystyle R}
, the semiperimeter
s
{\displaysThe following relations hold among the inradius
r
{\displaystyle r}
, the circumradius
R
{\displaystyle R}
, the semiperimeter
s
{\displaystyle s}
, and the excircle radii
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}
is the orthocenter of
△
A
B
C
{\displaystyle \triangle ABC}
, then[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}
is the orthocenter of
△
A
B
C
{\displaystyle \triangle ABC}
, then[13]
The Nagel triangle or extouch triangle of
△
A
B
C
{\displaystyle \triangle ABC}
is denoted by the vertices
T
A
{\displaystyle T_{A}}
,
T
B
{\displaystyle T_{B}}
, and
T
C
{\displaystyle T_{C}}
that are the three points where the excircles touch the reference
△
A
B
C
{\displaystyle \triangle ABC}
and where
T
A
{\displaystyle T_{A}}
is opposite of
A
{\displaystyle A}
, etc. This
△
T
A
T
B
T
C
{\displaystyle \triangle T_{A}T_{B}T_{C}}
is also known as the extouch triangle of
△
A
B
C
{\displaystyle \triangle ABC}
. The circumcircle of the extouch
△
T
A
T
B
T
C
{\displaystyle \triangle T_{A}T_{B}T_{C}}
is called the Mandart circle .[citation needed ]
The three lines
A
T
A
{\displaystyle AT_{A}}
,
B
T
B
{\displaystyle BT_{B}}
and
C
T
C
{\displaystyle CT_{C}}
are called the 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}}
(or 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}
with the segments
B
C
{\displaystyle BC}
,
C
A
{\displaystyle CA}
, and
A
B
{\displaystyle AB}
are the vertices of the 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)}
. The four circles described above are given equivalently by either of the two given equations:[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}
and
r
{\displaystyle r}
are the circumradius and inradius respectively, and
d
{\displaystyle d}
is the distance between the 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}}}
is the radius of one of the excircles, and
d
ex
{\displaystyle d_{\text{ex}}}
is the distance between the circumcenter and that excircle's center.[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 . #84, p. 121.
^ 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 96, March 2012, 161-165.
^ Altshiller-Court, Nathan. College Geometry , Dover Publications, 1980.
^ Posamentier, Alfred S., and Lehmann, Ingmar. The Secrets of Triangles , Prometheus Books, 2012.
^ 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 115, October 2008, 679-689: Theorem 4.1.
^
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 6 (2006), 57–70. 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}
and
r
{\displaystyle r}
are the circumradius and inradius respectively, and
d
{\displaystyle d}
is the distance between the 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}}}
is the radius of one of the excircles, and
d
ex
{\displaystyle d_{\text{ex}}}
is the distance between the circumcenter and that excircle's center.[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 ]