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In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is called the triangle's incenter.
An excircle or escribed circle 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.
The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle and the external bisectors of the other two. 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.:p. 182
Suppose has an incircle with radius r and center I. Let a be the length of BC, b the length of AC, and c the length of AB. Now, the incircle is tangent to AB at some point C′, and so is right. Thus the radius C'I is an altitude of . Therefore has base length c and height r, and so has area . Similarly, has area and has area . Since these three triangles decompose , we see that
where is the area of and is its semiperimeter.
For an alternative formula, consider . This is a right-angled triangle with one side equal to r and the other side equal to . The same is true for . The large triangle is composed of 6 such triangles and the total area is:
The radii in the excircles are called the exradii. Let the excircle at side AB touch at side AC extended at G, and let this excircle's radius be and its center be . Then is an altitude of , so has area . By a similar argument, has area and has area . Thus
So, by symmetry,
By the law of cosines, we have
Combining this with the identity , we have
But , and so
which is Heron's formula.
Combining this with , we have
From these formulas 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 formula yields:
The Gergonne triangle (of ABC) is defined by the 3 touchpoints of the incircle on the 3 sides. Those vertices are denoted as TA, etc. The point that TA denotes, lies opposite to A.
This Gergonne triangle TATBTC is also known as the contact triangle or intouch triangle of ABC.
The three lines ATA, BTB and CTC intersect in a single point called Gergonne point, denoted as Ge - X(7). The Gergonne point lies in the open orthocentroidal disk punctured at its own center, and could be any point therein.
Trilinear coordinates for the vertices of the intouch triangle are given by
Trilinear coordinates for the Gergonne point are given by
or, equivalently, by the Law of Sines,
The Nagel triangle of ABC is denoted by the vertices XA, XB and XC that are the three points where the excircles touch the reference triangle ABC and where XA is opposite of A, etc. This triangle XAXBXC is also known as the extouch triangle of ABC. The circumcircle of the extouch triangle XAXBXC is called the Mandart circle. The three lines AXA, BXB and CXC are called the splitters of the triangle; they each bisect the perimeter of the triangle, and they intersect in a single point, the triangle's Nagel point Na - X(8).
Trilinear coordinates for the vertices of the extouch triangle are given by
Trilinear coordinates for the Nagel point are given by
or, equivalently, by the Law of Sines,
It is the isotomic conjugate of the Gergonne point.
The points of intersection of the interior angle bisectors of ABC with the segments BC,CA,AB are the vertices of the incentral triangle.
Trilinear coordinates for the vertices of the incentral triangle are given by
Trilinear coordinates for the vertices of the excentral triangle are given by
Let x : y : z be a variable point in trilinear coordinates, and let u = cos2(A/2), v = cos2(B/2), w = cos2(C/2). The four circles described above are given equivalently by either of the two given equations::p. 210-215
Euler's theorem states that in a triangle:
where R and rin are the circumradius and inradius respectively, and d is the distance between the circumcenter and the incenter.
For excircles the equation is similar:
Suppose the tangency points of the incircle divide the sides into lengths of x and y, y and z, and z and x. Then the incircle has the radius
and the area of the triangle is
If the altitudes from sides of lengths a, b, and c are ha, hb, and hc then the inradius r is one-third of the harmonic mean of these altitudes, i.e.
The product of the incircle radius r and the circumcircle radius R of a triangle with sides a, b, and c is:p. 189, #298(d)
Some relations among the sides, incircle radius, and circumcircle radius are:
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.
Denoting the center of the incircle of triangle ABC as I, we have
The distance from any vertex to the incircle tangency on either adjacent side is half the sum of the vertex's adjacent sides minus half the opposite side. Thus for example for vertex B and adjacent tangencies TA and TC,
The incircle radius is no greater than one-ninth the sum of the altitudes.:p. 289
The squared distance from the incenter I to the circumcenter O is given by:p.232
The circular hull of the excircles is internally tangent to each of the excircles, and thus is an Apollonius circle. The radius of this Apollonius circle is where r is the incircle radius and s is the semiperimeter of the triangle.
The following relations hold among the inradius r, the circumradius R, the semiperimeter s, and the excircle radii r'a, rb, rc:
The circle through the centers of the three excircles has radius 2R.
Some (but not all) quadrilaterals have an incircle. These are called tangential quadrilaterals. Among their many properties perhaps the most important is that their opposite sides have equal sums. This is called the Pitot theorem.