Template:Short description Template:Redirect Template:Distinguish
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.<ref>Template:Harvtxt</ref>
An excircle or escribed circle<ref name="Altshiller-Court 1925 74">Template:Harvtxt</ref> of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides.<ref name="Altshiller-Court 1925 73">Template:Harvtxt</ref>
The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors.<ref name="Altshiller-Court 1925 73"/><ref>Template:Harvtxt</ref> The center of an excircle is the intersection of the internal bisector of one angle (at vertex Template:Mvar, for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex Template:Mvar, or the excenter of Template:Mvar.<ref name="Altshiller-Court 1925 73"/> 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.Template:Sfn
Incircle and IncenterEdit
Suppose <math>\triangle ABC</math> has an incircle with radius <math>r</math> and center <math>I</math>. Let <math>a</math> be the length of <math>\overline{BC}</math>, <math>b</math> the length of <math>\overline{AC}</math>, and <math>c</math> the length of <math>\overline{AB}</math>.
Also let <math>T_A</math>, <math>T_B</math>, and <math>T_C</math> be the touchpoints where the incircle touches <math>\overline{BC}</math>, <math>\overline{AC}</math>, and <math>\overline{AB}</math>.
IncenterEdit
The incenter is the point where the internal angle bisectors of <math>\angle ABC, \angle BCA, \text{ and } \angle BAC</math> meet.
Trilinear coordinatesEdit
The trilinear coordinates for a point in the triangle is the ratio of all the distances to the triangle sides. Because the incenter is the same distance from all sides of the triangle, the trilinear coordinates for the incenter are<ref name="etc">Encyclopedia of Triangle Centers Template:Webarchive, accessed 2014-10-28.</ref>
- <math display=block>\ 1 : 1 : 1.</math>
Barycentric coordinatesEdit
The barycentric coordinates for a point in a triangle give weights such that the point is the weighted average of the triangle vertex positions. Barycentric coordinates for the incenter are given by
- <math display=block>a : b : c</math>
where <math>a</math>, <math>b</math>, and <math>c</math> are the lengths of the sides of the triangle, or equivalently (using the law of sines) by
- <math display=block>\sin A : \sin B : \sin C</math>
where <math>A</math>, <math>B</math>, and <math>C</math> are the angles at the three vertices.
Cartesian coordinatesEdit
The Cartesian coordinates of the incenter are a weighted average of the coordinates of the three vertices using the side lengths of the triangle relative to the perimeter (that is, using the barycentric coordinates given above, normalized to sum to unity) as weights. The weights are positive so the incenter lies inside the triangle as stated above. If the three vertices are located at <math>(x_a,y_a)</math>, <math>(x_b,y_b)</math>, and <math>(x_c,y_c)</math>, and the sides opposite these vertices have corresponding lengths <math>a</math>, <math>b</math>, and <math>c</math>, then the incenter is atTemplate:Citation needed
- <math display=block>
\left(\frac{a x_a + b x_b + c x_c}{a + b + c}, \frac{a y_a + b y_b + c y_c}{a + b + c}\right)
= \frac{a\left(x_a, y_a\right) + b\left(x_b, y_b\right) + c\left(x_c, y_c\right)}{a + b + c}.
</math>
RadiusEdit
The inradius <math>r</math> of the incircle in a triangle with sides of length <math>a</math>, <math>b</math>, <math>c</math> is given by<ref>Template:Harvtxt</ref>
- <math display=block>r = \sqrt{\frac{(s-a)(s-b)(s-c)}{s}},</math>
where <math>s = \tfrac12(a + b + c)</math> is the semiperimeter (see Heron's formula).
The tangency points of the incircle divide the sides into segments of lengths <math>s-a</math> from <math>A</math>, <math>s-b</math> from <math>B</math>, and <math>s-c</math> from <math>C</math> (see Tangent lines to a circle).<ref>Chu, Thomas, The Pentagon, Spring 2005, p. 45, problem 584.</ref>
Distances to the verticesEdit
Denote the incenter of <math>\triangle ABC</math> as <math>I</math>.
The distance from vertex <math>A</math> to the incenter <math>I</math> is:
- <math display=block>
\overline{AI} = d(A, I)
= c \, \frac{\sin\frac{B}{2}}{\cos\frac{C}{2}}
= b \, \frac{\sin\frac{C}{2}}{\cos\frac{B}{2}}.
</math>
Derivation of the formula stated aboveEdit
Use the Law of sines in the triangle <math>\triangle IAB</math>.
We get <math>\frac{\overline{AI}}{\sin \frac{B}{2}} = \frac{c}{\sin \angle AIB}</math>. We have that <math>\angle AIB = \pi - \frac{A}{2} - \frac{B}{2} = \frac{\pi}{2} + \frac{C}{2}</math>.
It follows that <math>\overline{AI} = c \ \frac{\sin \frac{B}{2}}{\cos \frac{C}{2}}</math>.
The equality with the second expression is obtained the same way.
The distances from the incenter to the vertices combined with the lengths of the triangle sides obey the equation<ref> Template:Citation.</ref>
- <math display=block>\frac{\overline{IA} \cdot \overline{IA}}{\overline{CA} \cdot \overline{AB}} + \frac{\overline{IB} \cdot \overline{IB}}{\overline{AB} \cdot \overline{BC}} + \frac{\overline{IC} \cdot \overline{IC}}{\overline{BC} \cdot \overline{CA}} = 1.</math>
Additionally,<ref>Template:Citation. #84, p. 121.</ref>
- <math display=block>\overline{IA} \cdot \overline{IB} \cdot \overline{IC} = 4Rr^2,</math>
where <math>R</math> and <math>r</math> are the triangle's circumradius and inradius respectively.
Other propertiesEdit
The collection of triangle centers may be given the structure of a group under coordinate-wise multiplication of trilinear coordinates; in this group, the incenter forms the identity element.<ref name="etc"/>
Incircle and its radius propertiesEdit
Distances between vertex and nearest touchpointsEdit
The distances from a vertex to the two nearest touchpoints are equal; for example:<ref name=":0">Mathematical Gazette, July 2003, 323-324.</ref>
- <math display=block>d\left(A, T_B\right) = d\left(A, T_C\right) = \tfrac12(b + c - a) = s - a.</math>
Other propertiesEdit
If the altitudes from sides of lengths <math>a</math>, <math>b</math>, and <math>c</math> are <math>h_a</math>, <math>h_b</math>, and <math>h_c</math>, then the inradius <math>r</math> is one-third of the harmonic mean of these altitudes; that is,<ref>Template:Harvtxt</ref>
- <math display=block> r = \frac{1}{\dfrac{1}{h_a} + \dfrac{1}{h_b} + \dfrac{1}{h_c}}.</math>
The product of the incircle radius <math>r</math> and the circumcircle radius <math>R</math> of a triangle with sides <math>a</math>, <math>b</math>, and <math>c</math> isTemplate:Sfn
- <math display=block>rR = \frac{abc}{2(a + b + c)}.</math>
Some relations among the sides, incircle radius, and circumcircle radius are:<ref name=Bell/>
- <math display=block>\begin{align}
ab + bc + ca &= s^2 + (4R + r)r, \\ a^2 + b^2 + c^2 &= 2s^2 - 2(4R + r)r.
\end{align}</math>
Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its incircle). There are either one, two, or three of these for any given triangle.<ref>Kodokostas, Dimitrios, "Triangle Equalizers", Mathematics Magazine 83, April 2010, pp. 141-146.</ref>
The incircle radius is no greater than one-ninth the sum of the altitudes.<ref>Posamentier, Alfred S., and Lehmann, Ingmar. The Secrets of Triangles, Prometheus Books, 2012.</ref>Template:Rp
The squared distance from the incenter <math>I</math> to the circumcenter <math>O</math> is given by<ref name=Franzsen>Template:Cite journal.</ref>Template:Rp
- <math display=block>\overline{OI}^2 = R(R - 2r) = \frac{a\,b\,c\,}{a+b+c}\left [\frac{a\,b\,c\,}{(a+b-c)\,(a-b+c)\,(-a+b+c)}-1 \right ]</math>
and the distance from the incenter to the center <math>N</math> of the nine point circle is<ref name=Franzsen/>Template:Rp
- <math display=block>\overline{IN} = \tfrac12(R - 2r) < \tfrac12 R.</math>
The incenter lies in the medial triangle (whose vertices are the midpoints of the sides).<ref name=Franzsen/>Template:Rp
Relation to area of the triangleEdit
Template:Redirect The radius of the incircle is related to the area of the triangle.<ref>Coxeter, H.S.M. "Introduction to Geometry 2nd ed. Wiley, 1961.</ref> The ratio of the area of the incircle to the area of the triangle is less than or equal to <math>\pi \big/ 3\sqrt3</math>, with equality holding only for equilateral triangles.<ref>Minda, D., and Phelps, S., "Triangles, ellipses, and cubic polynomials", American Mathematical Monthly 115, October 2008, 679-689: Theorem 4.1.</ref>
Suppose <math>\triangle ABC</math> has an incircle with radius <math>r</math> and center <math>I</math>. Let <math>a</math> be the length of <math>\overline{BC}</math>, <math>b</math> the length of <math>\overline{AC}</math>, and <math>c</math> the length of <math>\overline{AB}</math>.
Now, the incircle is tangent to <math>\overline{AB}</math> at some point <math>T_C</math>, and so <math>\angle AT_CI</math> is right. Thus, the radius <math>T_CI</math> is an altitude of <math>\triangle IAB</math>.
Therefore, <math>\triangle IAB</math> has base length <math>c</math> and height <math>r</math>, and so has area <math>\tfrac12 cr</math>.
Similarly, <math>\triangle IAC</math> has area <math>\tfrac12 br</math> and <math>\triangle IBC</math> has area <math>\tfrac12 ar</math>.
Since these three triangles decompose <math>\triangle ABC</math>, we see that the area <math>\Delta \text{ of} \triangle ABC</math> is:
- <math display=block>\Delta = \tfrac12 (a + b + c)r = sr,</math>
Template:Spaces and Template:Spaces<math>r = \frac{\Delta}{s},</math>
where <math>\Delta</math> is the area of <math>\triangle ABC</math> and <math>s = \tfrac12(a + b + c)</math> is its semiperimeter.
For an alternative formula, consider <math>\triangle IT_CA</math>. This is a right-angled triangle with one side equal to <math>r</math> and the other side equal to <math>r \cot \tfrac{A}{2}</math>. The same is true for <math>\triangle IB'A</math>. The large triangle is composed of six such triangles and the total area is:Template:Citation needed
- <math display=block>\Delta = r^2 \left(\cot\tfrac{A}{2} + \cot\tfrac{B}{2} + \cot\tfrac{C}{2}\right).</math>
Gergonne triangle and pointEdit
The Gergonne triangle (of <math>\triangle ABC</math>) is defined by the three touchpoints of the incircle on the three sides. The touchpoint opposite <math>A</math> is denoted <math>T_A</math>, etc.
This Gergonne triangle, <math>\triangle T_AT_BT_C</math>, is also known as the contact triangle or intouch triangle of <math>\triangle ABC</math>. Its area is
- <math display=block>K_T = K\frac{2r^2 s}{abc}</math>
where <math>K</math>, <math>r</math>, and <math>s</math> are the area, radius of the incircle, and semiperimeter of the original triangle, and <math>a</math>, <math>b</math>, and <math>c</math> are the side lengths of the original triangle. This is the same area as that of the extouch triangle.<ref> Weisstein, Eric W. "Contact Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ContactTriangle.html</ref>
The three lines <math>AT_A</math>, <math>BT_B</math>, and <math>CT_C</math> intersect in a single point called the Gergonne point, denoted as <math>G_e</math> (or triangle center X7). The Gergonne point lies in the open orthocentroidal disk punctured at its own center, and can be any point therein.<ref name=Bradley>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</ref>
The Gergonne point of a triangle has a number of properties, including that it is the symmedian point of the Gergonne triangle.<ref> Template:Cite journal</ref>
Trilinear coordinates for the vertices of the intouch triangle are given byTemplate:Citation needed
- <math display=block>\begin{array}{ccccccc}
T_A &=& 0 &:& \sec^2 \frac{B}{2} &:& \sec^2\frac{C}{2} \\[2pt]
T_B &=& \sec^2 \frac{A}{2} &:& 0 &:& \sec^2\frac{C}{2} \\[2pt]
T_C &=& \sec^2 \frac{A}{2} &:& \sec^2\frac{B}{2} &:& 0.
\end{array}</math>
Trilinear coordinates for the Gergonne point are given byTemplate:Citation needed
- <math display=block>\sec^2\tfrac{A}{2} : \sec^2\tfrac{B}{2} : \sec^2\tfrac{C}{2},</math>
or, equivalently, by the Law of Sines,
- <math display=block>\frac{bc}{b + c - a} : \frac{ca}{c + a - b} : \frac{ab}{a + b - c}.</math>
Excircles and excentersEdit
An excircle or escribed circle<ref name="Altshiller-Court 1925 74"/> of the triangle is a circle lying outside the triangle, tangent to one of its sides, and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides.<ref name="Altshiller-Court 1925 73"/>
The center of an excircle is the intersection of the internal bisector of one angle (at vertex <math>A</math>, for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex <math>A</math>, or the excenter of <math>A</math>.<ref name="Altshiller-Court 1925 73"/> 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.Template:Sfn
Trilinear coordinates of excentersEdit
While the incenter of <math>\triangle ABC</math> has trilinear coordinates <math>1 : 1 : 1</math>, the excenters have trilinears Template:Citation needed
- <math display=block>\begin{array}{rrcrcr}
J_A = & -1 &:& 1 &:& 1 \\ J_B = & 1 &:& -1 &:& 1 \\ J_C = & 1 &:& 1 &:& -1
\end{array}</math>
ExradiiEdit
The radii of the excircles are called the exradii.
The exradius of the excircle opposite <math>A</math> (so touching <math>BC</math>, centered at <math>J_A</math>) is<ref name="Altshiller-Court 1925 79">Template:Harvtxt</ref><ref>Template:Harvtxt</ref>
- <math display=block>r_a = \frac{rs}{s - a} = \sqrt{\frac{s(s - b)(s - c)}{s - a}},</math> where <math>s = \tfrac{1}{2}(a + b + c).</math>
See Heron's formula.
Derivation of exradii formulaEdit
Source:<ref name="Altshiller-Court 1925 79"/>
Let the excircle at side <math>AB</math> touch at side <math>AC</math> extended at <math>G</math>, and let this excircle's radius be <math>r_c</math> and its center be <math>J_c</math>. Then <math>J_c G</math> is an altitude of <math>\triangle ACJ_c</math>, so <math>\triangle ACJ_c</math> has area <math>\tfrac12 br_c</math>. By a similar argument, <math>\triangle BCJ_c</math> has area <math>\tfrac12 ar_c</math> and <math>\triangle ABJ_c</math> has area <math>\tfrac12 cr_c</math>. Thus the area <math>\Delta</math> of triangle <math>\triangle ABC</math> is
- <math display=block>\Delta = \tfrac12 (a + b - c)r_c = (s - c)r_c</math>.
So, by symmetry, denoting <math>r</math> as the radius of the incircle,
- <math display=block>\Delta = sr = (s - a)r_a = (s - b)r_b = (s - c)r_c</math>.
By the Law of Cosines, we have
- <math display=block>\cos A = \frac{b^2 + c^2 - a^2}{2bc}</math>
Combining this with the identity <math>\sin^2 \! A + \cos^2 \! A = 1</math>, we have
- <math display=block>\sin A = \frac{\sqrt{-a^4 - b^4 - c^4 + 2a^2 b^2 + 2b^2 c^2 + 2 a^2 c^2}}{2bc}</math>
But <math>\Delta = \tfrac12 bc \sin A</math>, and so
- <math display=block>\begin{align}
\Delta &= \tfrac14 \sqrt{-a^4 - b^4 - c^4 + 2a^2b^2 + 2b^2 c^2 + 2 a^2 c^2} \\[5mu]
&= \tfrac14 \sqrt{(a + b + c)(-a + b + c)(a - b + c)(a + b - c)} \\[5mu]
& = \sqrt{s(s - a)(s - b)(s - c)},
\end{align}</math>
which is Heron's formula.
Combining this with <math>sr = \Delta</math>, we have
- <math display=block>r^2 = \frac{\Delta^2}{s^2} = \frac{(s - a)(s - b)(s - c)}{s}.</math>
Similarly, <math>(s - a)r_a = \Delta</math> gives
- <math display=block>\begin{align}
&r_a^2 = \frac{s(s - b)(s - c)}{s - a} \\[4pt]
&\implies r_a = \sqrt{\frac{s(s - b)(s - c)}{s - a}}.
\end{align}</math>
Other propertiesEdit
From the formulas above one can see that the excircles are always larger than the incircle and that the largest excircle is the one tangent to the longest side and the smallest excircle is tangent to the shortest side. Further, combining these formulas yields:<ref>Baker, Marcus, "A collection of formulae for the area of a plane triangle", Annals of Mathematics, part 1 in vol. 1(6), January 1885, 134-138. (See also part 2 in vol. 2(1), September 1885, 11-18.)</ref>
- <math display=block>\Delta = \sqrt{r r_a r_b r_c}.</math>
Other excircle propertiesEdit
The circular hull of the excircles is internally tangent to each of the excircles and is thus an Apollonius circle.<ref>Grinberg, Darij, and Yiu, Paul, "The Apollonius Circle as a Tucker Circle", Forum Geometricorum 2, 2002: pp. 175-182.</ref> The radius of this Apollonius circle is <math>\tfrac{r^2 + s^2}{4r}</math> where <math>r</math> is the incircle radius and <math>s</math> is the semiperimeter of the triangle.<ref>Stevanovi´c, Milorad R., "The Apollonius circle and related triangle centers", Forum Geometricorum 3, 2003, 187-195.</ref>
The following relations hold among the inradius <math>r</math>, the circumradius <math>R</math>, the semiperimeter <math>s</math>, and the excircle radii <math>r_a</math>, <math>r_b</math>, <math>r_c</math>:<ref name=Bell>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
- <math display=block>\begin{align}
r_a + r_b + r_c &= 4R + r, \\
r_a r_b + r_b r_c + r_c r_a &= s^2, \\
r_a^2 + r_b^2 + r_c^2 &= \left(4R + r\right)^2 - 2s^2.
\end{align}</math>
The circle through the centers of the three excircles has radius <math>2R</math>.<ref name=Bell/>
If <math>H</math> is the orthocenter of <math>\triangle ABC</math>, then<ref name=Bell/>
- <math display=block>\begin{align}
r_a + r_b + r_c + r &= \overline{AH} + \overline{BH} + \overline{CH} + 2R, \\
r_a^2 + r_b^2 + r_c^2 + r^2 &= \overline{AH}^2 + \overline{BH}^2 + \overline{CH}^2 + (2R)^2.
\end{align}</math>
Nagel triangle and Nagel pointEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}
The Nagel triangle or extouch triangle of <math>\triangle ABC</math> is denoted by the vertices <math>T_A</math>, <math>T_B</math>, and <math>T_C</math> that are the three points where the excircles touch the reference <math>\triangle ABC</math> and where <math>T_A</math> is opposite of <math>A</math>, etc. This <math>\triangle T_AT_BT_C</math> is also known as the extouch triangle of <math>\triangle ABC</math>. The circumcircle of the extouch <math>\triangle T_AT_BT_C</math> is called the Mandart circle (cf. Mandart inellipse).
The three line segments <math>\overline{AT_A}</math>, <math>\overline{BT_B}</math> and <math>\overline{CT_C}</math> are called the splitters of the triangle; they each bisect the perimeter of the triangle,Template:Citation needed
- <math display=block>\overline{AB} + \overline{BT_A} = \overline{AC} + \overline{CT_A} = \frac{1}{2}\left( \overline{AB} + \overline{BC} + \overline{AC} \right).</math>
The splitters intersect in a single point, the triangle's Nagel point <math>N_a</math> (or triangle center X8).
Trilinear coordinates for the vertices of the extouch triangle are given byTemplate:Citation needed
- <math display=block>\begin{array}{ccccccc}
T_A &=& 0 &:& \csc^2\frac{B}{2} &:& \csc^2\frac{C}{2} \\[2pt]
T_B &=& \csc^2\frac{A}{2} &:& 0 &:& \csc^2\frac{C}{2} \\[2pt]
T_C &=& \csc^2\frac{A}{2} &:& \csc^2\frac{B}{2} &:& 0
\end{array}</math>
Trilinear coordinates for the Nagel point are given byTemplate:Citation needed
- <math display=block>\csc^2\tfrac{A}{2} : \csc^2\tfrac{B}{2} : \csc^2\tfrac{C}{2},</math>
or, equivalently, by the Law of Sines,
- <math display=block>\frac{b + c - a}{a} : \frac{c + a - b}{b} : \frac{a + b - c}{c}.</math>
The Nagel point is the isotomic conjugate of the Gergonne point.Template:Citation needed
Related constructionsEdit
Nine-point circle and Feuerbach pointEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}
In 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:<ref>Template:Harvtxt</ref><ref>Template:Harvtxt</ref>
- The midpoint of each side of the triangle
- The foot of each altitude
- The midpoint of the line segment from each vertex of the triangle to the orthocenter (where the three altitudes meet; these line segments lie on their respective altitudes).
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:<ref>Template:Citation.</ref>
- ... 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 ... Template:Harv
The triangle center at which the incircle and the nine-point circle touch is called the Feuerbach point.
Incentral and excentral trianglesEdit
The points of intersection of the interior angle bisectors of <math>\triangle ABC</math> with the segments <math>BC</math>, <math>CA</math>, and <math>AB</math> are the vertices of the incentral triangle. Trilinear coordinates for the vertices of the incentral triangle <math>\triangle A'B'C'</math> are given byTemplate:Citation needed
- <math display=block>\begin{array}{ccccccc}
A' &=& 0 &:& 1 &:& 1 \\[2pt] B' &=& 1 &:& 0 &:& 1 \\[2pt] C' &=& 1 &:& 1 &:& 0
\end{array}</math>
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 <math>\triangle A'B'C'</math> are given byTemplate:Citation needed
- <math display=block>\begin{array}{ccrcrcr}
A' &=& -1 &:& 1 &:& 1\\[2pt] B' &=& 1 &:& -1 &:& 1 \\[2pt] C' &=& 1 &:& 1 &:& -1
\end{array}</math>
Equations for four circlesEdit
Let <math>x:y:z</math> be a variable point in trilinear coordinates, and let <math>u=\cos^2\left ( A/2 \right )</math>, <math>v=\cos^2\left ( B/2 \right )</math>, <math>w=\cos^2\left ( C/2 \right )</math>. The four circles described above are given equivalently by either of the two given equations:<ref name=WW>Whitworth, William Allen. Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions, Forgotten Books, 2012 (orig. Deighton, Bell, and Co., 1866). https://www.forgottenbooks.com/en/search?q=%22Trilinear+coordinates%22</ref>Template:Rp
- Incircle:<math display=block>\begin{align}
u^2 x^2 + v^2 y^2 + w^2 z^2 - 2vwyz - 2wuzx - 2uvxy &= 0 \\[4pt]
{\textstyle \pm\sqrt{x}\cos\tfrac{A}{2} \pm \sqrt{y\vphantom{t}}\cos\tfrac{B}{2} \pm \sqrt{z}\cos\tfrac{C}{2}} &= 0
\end{align}</math>
- <math>A</math>-excircle:<math display=block>\begin{align}
u^2 x^2 + v^2 y^2 + w^2 z^2 - 2vwyz + 2wuzx + 2uvxy &= 0 \\[4pt]
{\textstyle \pm\sqrt{-x}\cos\tfrac{A}{2} \pm \sqrt{y\vphantom{t}}\cos\tfrac{B}{2} \pm \sqrt{z}\cos\tfrac{C}{2}} &= 0
\end{align}</math>
- <math>B</math>-excircle:<math display=block>\begin{align}
u^2 x^2 + v^2 y^2 + w^2 z^2 + 2vwyz - 2wuzx + 2uvxy &= 0 \\[4pt]
{\textstyle \pm\sqrt{x}\cos\tfrac{A}{2} \pm \sqrt{-y\vphantom{t}}\cos\tfrac{B}{2} \pm \sqrt{z}\cos\tfrac{C}{2}} &= 0
\end{align}</math>
- <math>C</math>-excircle:<math display=block>\begin{align}
u^2 x^2 + v^2 y^2 + w^2 z^2 + 2vwyz + 2wuzx - 2uvxy &= 0 \\[4pt]
{\textstyle \pm\sqrt{x}\cos\tfrac{A}{2} \pm \sqrt{y\vphantom{t}}\cos\tfrac{B}{2} \pm \sqrt{-z}\cos\tfrac{C}{2}} &= 0
\end{align}</math>
Euler's theoremEdit
Euler's theorem states that in a triangle:
- <math display=block>(R - r)^2 = d^2 + r^2,</math>
where <math>R</math> and <math>r</math> are the circumradius and inradius respectively, and <math>d</math> is the distance between the circumcenter and the incenter.
For excircles the equation is similar:
- <math display=block>\left(R + r_\text{ex}\right)^2 = d_\text{ex}^2 + r_\text{ex}^2,</math>
where <math>r_\text{ex}</math> is the radius of one of the excircles, and <math>d_\text{ex}</math> is the distance between the circumcenter and that excircle's center.<ref name=Nelson>Nelson, Roger, "Euler's triangle inequality via proof without words", Mathematics Magazine 81(1), February 2008, 58-61.</ref>Template:Sfn<ref>Emelyanov, Lev, and Emelyanova, Tatiana. "Euler's formula and Poncelet's porism", Forum Geometricorum 1, 2001: pp. 137–140.</ref>
Generalization to other polygonsEdit
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.<ref>Template:Harvtxt</ref>
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.
See alsoEdit
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- Triangle conic
NotesEdit
ReferencesEdit
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External linksEdit
- Derivation of formula for radius of incircle of a triangle, MATHalino
- {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web
|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Incircle%7CIncircle.html}} |title = Incircle |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}
InteractiveEdit
- Triangle incenter; Triangle incircle; Incircle of a regular polygon (with interactive animations)
- Constructing a triangle's incenter / incircle with compass and straightedge An interactive animated demonstration
- Equal Incircles Theorem at cut-the-knot
- Five Incircles Theorem at cut-the-knot
- Pairs of Incircles in a Quadrilateral at cut-the-knot
- An interactive Java applet for the incenter