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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...

{\displaystyle a} and b {\displaystyle b} are the legs of the triangle. If the incircle is tangent to the hypotenuse A B {\displaystyle AB} at point P , {\displaystyle...

mixtilinear incircle of a triangle is a circle which is tangent to two of its sides and internally tangent to its circumcircle. The mixtilinear incircle of a...

the only triangle whose Steiner inellipse is a circle (specifically, the incircle). The triangle of the largest area of all those inscribed in a given circle...

tangent to that triangle's three excircles and internally tangent to its incircle; this result is known as Feuerbach's theorem. He proved that: ... the circle...

single point, the incenter, which is the center of the triangle's incircle. The incircle is the largest circle that lies inside the triangle; it touches...

to a single circle within the quadrilateral. This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the incenter...

quadrilaterals (quadrilaterals with both a circumcircle and an incircle), since all kites have an incircle. One of the diagonals (the one that is a line of symmetry)...

every tangential quadrilateral other than a rhombus, the center of the incircle lies on the Newton line. Let ABCD be a tangential quadrilateral with at...

whose four sides are all tangent to a circle within the trapezoid: the incircle or inscribed circle. It is the special case of a tangential quadrilateral...

In geometry, the equal incircles theorem derives from a Japanese Sangaku, and pertains to the following construction: a series of rays are drawn from...

the incenter only exists for tangential polygons: those that have an incircle that is tangent to each side of the polygon. In this case the incenter...

circle of infinite radius. In every triangle a unique circle, called the incircle, can be inscribed such that it is tangent to each of the three sides of...

a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and centers of these circles are called inradius...

In the geometry of triangles, the incircle and nine-point circle of a non-equilateral triangle are internally tangent to each other at the Feuerbach point...

triangle △ABC   Congruent isoscelizers of △ABC   Incircle of △ABC   Intouch triangle △A'B'C' of △ABC   Incircle of △A'B'C' (used to construct △A"B"C")   Perspective...

As a corollary of the chord formula, the area bounded by the circumcircle and incircle of every unit convex regular polygon is π/4...

circumcircle. Its centre is equidistant from its sides, hence it has an incircle. Two axes of symmetry bisect opposite sides. Two axes of symmetry bisect...

software engineer who developed the social networking services Club Nexus, inCircle and Orkut. He is a former product manager at Google. Originally from Konya...

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....

As a corollary of the annulus chord formula, the area bounded by the circumcircle and incircle of every unit regular n-gon is π/4...

this implies that every regular polygon also has an inscribed circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is...

{(s-a)(s-b)(s-c)}{s}}}\,.} In the upper figure, the points of tangency of the incircle with the sides of the triangle break the perimeter into 6 segments, in...

tangential polygon (a polygon all of whose sides are tangent to an inner incircle) which is also cyclic — that is, inscribed in an outer circle that passes...

inscribed in circles. A circle inscribed in any polygon is called its incircle, in which case the polygon is said to be a tangential polygon. A polygon...

of the hyperbola is the Feuerbach point, the point of tangency of the incircle and the nine-point circle. It has the trilinear equation cos ⁡ B − cos...

sides tangent to a particular circle, called the incircle or inscribed circle. The centre of the incircle, called the incentre, can be considered a centre...

that touches the three sides of a triangle. The simplest example is the incircle. Further important inellipses are the Steiner inellipse, which touches...

concentric circles (with that distance being zero) are the circumcircle and incircle of a triangle if and only if the radius of one is twice the radius of the...

1809. L'Huilier's theorem—Let r {\displaystyle r} be the radius of the incircle of a triangle and r A , r B , r C {\displaystyle r_{\text{A}},r_{\text{B}}...

is a convex polygon that contains an inscribed circle (also called an incircle). This is a circle that is tangent to each of the polygon's sides. The...

Dimensions related to an ideal triangle and its incircle, depicted in the Beltrami–Klein model (left) and the Poincaré disk model (right)...

as its circumcircle and incircle, there exist an infinite number of other triangles with the same circumcircle and incircle, with any point on the circumcircle...

In geometry, the Japanese theorem states that the centers of the incircles of certain triangles inside a cyclic quadrilateral are vertices of a rectangle...

Two triangles are said to be poristic triangles if they have the same incircle and circumcircle. Given a circle with center O and radius R and another...

previously working for Affinity Engines, he had developed a similar system, InCircle, intended for use by university alumni groups. In late June 2004, Affinity...

Let I be the center of the incircle of triangle ABC, r its radius and Fa, Fb and Fc the three points where the incircle touches the triangle sides a...

concur at (all pass through) the center of the Spieker circle, which is the incircle of the medial triangle. The cleavers are parallel to the angle bisectors...

Graeco-Latin squares Euler's theorem in geometry, relating the circumcircle and incircle of a triangle Euler's quadrilateral theorem, an extension of the parallelogram...

{1}{2}}(a+b+c)} ⁠, and ⁠ r {\displaystyle r} ⁠ is the radius of the triangle's incircle, then ⁠ r s {\displaystyle rs} ⁠ is the triangle's area. Therefore Heron's...

In geometry, the incircle of the medial triangle of a triangle is the Spieker circle, named after 19th-century German geometer Theodor Spieker. Its center...

through a vertex and perpendicular to any line tangent to the triangle's incircle. In a square or other rectangle, all pairs of adjacent sides are perpendicular...

Carnot's theorem (inradius, circumradius), describing a property of the incircle and the circumcircle of a triangle Carnot's theorem (conics), describing...

equilateral. Given any triangle ABC, and any point M on BC, construct the incircle and circumcircle of the triangle. Then construct two additional circles...

circles Bankoff circle Circular triangle Reuleaux triangle Circumcircle Disc Incircle and excircles of a triangle Nine-point circle Circular sector Circular...

Focus Circle List of circle topics Thales' theorem Circumcircle Concyclic Incircle and excircles of a triangle Orthocentric system Monge's theorem Power center...

BCP, CDP, and DAP are all congruent a quadrilateral ABCD in which the incircles in triangles ABC, BCD, CDA and DAB have a common point Every rhombus has...

polyedron at the center of the incircle of that face. If the exsphere radius is denoted rex, the radius of this incircle rin and the dihedral angle between...

for hyperbolic tangent and cotangent). e (mathematical constant) Equal incircles theorem, based on sinh Hyperbolastic functions Hyperbolic growth Inverse...

  Triangle △ABC (side lengths a, b, c)   Incircle (centered at incenter I)   Isoperimetric lines dA, dB, dC (concur at isoperimetric point Q)   Detours...

triangle, where its incircle is tangent to the side. The two circles tangent to these three circles are separated by the incircle, one interior to it...

isometric with incircles of diameter 2 log 3. Note that in this case the Gromov product also has a simple interpretation in terms of the incircle of a geodesic...

in a line called a cleaver of the triangle. (The Spieker circle is the incircle of the medial triangle, and its center is the center of mass of the perimeter...

lengths, including the diameter of the circumcircle, the radius of the incircle, and the radii of the three excircles, are also positive integers. Moreover...

c and with corresponding altitudes ha, hb, hc. The altitudes and the incircle radius r are related by 1 r = 1 h a + 1 h b + 1 h c . {\displaystyle \displaystyle...

an equilateral triangle: a) To any equilateral triangle there exists an incircle. It touches the sides at its midpoints. There is no other (non-degenerate)...

points are equal (this is sometimes called the Two Tangents Theorem, see Incircle). By the secant-tangent theorem, the square of this tangent length equals...

three cleavers concur at the center of the Spieker circle, which is the incircle of the medial triangle; the Spieker center is the center of mass of all...

perpendicular distances of its vertices from an arbitrary line tangent to its incircle. The theorem is named after J. Harcourt, an Irish professor. Let a triangle...

triangle. Here r {\displaystyle r} denotes the radius of the triangles incircle. Roger A. Johnson: Advanced Euclidean Geometry. Dover 2007, ISBN 978-0-486-46237-0...

compared to the area of the incircle: Area of incircle Area of triangle ≤ π 3 3 {\displaystyle {\frac {\text{Area of incircle}}{\text{Area of triangle}}}\leq...

inequality is an inequality relating the length, the area, the radius of the incircle and the radius of the circumcircle of a Jordan curve. It is a strengthening...

real numbers, it then follows that the integral taken over the square's incircle must be less than I ( a ) 2 {\displaystyle I(a)^{2}} , and similarly the...

{s(s-a)(s-b)(s-c)}}.\end{aligned}}} If ⁠ r {\displaystyle r} ⁠ is the radius of the incircle of the triangle, then the triangle can be broken into three triangles of...

three cleavers concur at the center of the Spieker circle, which is the incircle of the medial triangle. A splitter of a triangle is a line segment having...

determine the shape of the pentagon. A tangential polygon (one that has an incircle tangent to all its sides) is equilateral if and only if the alternate angles...

Pythagorean theorem. A tangential trapezoid is a trapezoid that has an incircle. Four lengths a, c, b, d can constitute the consecutive sides of a non-parallelogram...

{\displaystyle ABC} , and let I {\displaystyle I} be the center of its incircle. If the ray A I {\displaystyle AI} intersects the circumcircle at the point...

length. The products of opposite side lengths are equal. The center of the incircle lies on a line of symmetry that is also a diagonal. If the diagonals in...

{\displaystyle P} is the incenter of the triangle then the pedal circle is the incircle of the triangle and if P {\displaystyle P} is the orthocenter of the triangle...

reciprocals of the altitudes equals the reciprocal of the radius of the incircle (regardless of whether or not they are integers). In a right triangle,...

for a roundtrip journey (see above). In any triangle, the radius of the incircle is one-third of the harmonic mean of the altitudes. For any point P on...

triangle The triangle whose vertices are the points of contact of the incircle with the sides of ABC Tangential triangle Extouch triangle The triangle...

{CH}}^{2}=12R^{2}.} In addition, denoting r as the radius of the triangle's incircle, ra, rb, rc as the radii of its excircles, and R again as the radius of...

this maximum. As in Euclidean geometry, each hyperbolic triangle has an incircle. In hyperbolic space, if all three of its vertices lie on a horocycle or...

as can be seen in the table below. Thus a convex quadrilateral has an incircle or an excircle outside the appropriate vertex (depending on the column)...

ETC reference Name Definition X(1) Incenter center of the incircle X(2) Centroid intersection of the three medians X(3) Circumcenter center of the circumscribed...

important triangle centers.   Reference triangle △ABC   Angle bisectors and incircle (intersect/centered at incenter I)   Medians (intersect at centroid G)...

b, c], its area A, and its perimeter P, as well as the radii ri of its incircle and three excircles as follows: a = 2 q p b = q ′ p ′ c = p p ′ − q q ′...

associated with it in some way. For example, the circumcircle and the incircle of the reference triangle are triangle conics. Other examples are the Steiner...

triangle's circumcircle at the reference triangle's vertices. Thus the incircle of the tangential triangle coincides with the circumcircle of the reference...

triangle is equilateral, the Steiner inellipse is simply the triangle's incircle, its foci coincide with each other at the incenter, which lies on the real...

theorem, Apollonius' theorem, British flag theorem, Ceva's theorem, Equal incircles theorem, Geometric mean theorem, Heron's formula, Isosceles triangle theorem...

triangle, the discovery of Poncelet's porism on triangles with a common incircle and circumcircle. He was also one of the earliest mathematicians to calculate...

rolling circle. The perimeter (total arc length) of the deltoid is 16a. The incircle of the deltoid has radius R/2, this means 3/2 a. Ordinary cycloids were...

sedans for early 2007, available only through the Neiman Marcus catalog for InCircle members. These special models came in an exclusive Truffle Mica color with...

isolate Britain, and strengthen Germany’s position breaking out isolation incircle before the 20th-century balance of power. France's involvement in the Anglo–Boer...

product (A, B)C equals the distance p between C and the point where the incircle of the geodesic triangle ABC touches the edge CB or CA. Indeed from the...

The circle H touches the incircles I, I A {\displaystyle I_{A}} , I B {\displaystyle I_{B}} , I C {\displaystyle I_{C}} of a circular triangle ABC and...

in geometry are named after him: one that describes a property of the incircle and the circumcircle of a triangle, one that describes a relation between...

body-center. The 35 lines correspond to the 6 edges + 12 face-medians + 4 face-incircles + 4 altitudes from a face to the opposite vertex + 3 lines connecting...

for an obtuse triangle. For an acute triangle the distance between the incircle center I and orthocenter H satisfies I H < r 2 , {\displaystyle IH<r{\sqrt...

a circumgonal region because it circumscribes the circle known as the incircle of the triangle. Every square is a circumgonal region. In fact, every regular...

theorem (plane geometry) Dinostratus' theorem (geometry, analysis) Equal incircles theorem (Euclidean geometry) Euler's quadrilateral theorem (geometry)...

siege by digging a line that was only a semicircle and did not completely incircle the city. The attack was concentrated between the Santa Cruz and São João...

its OI line. The proof roughly goes as below: Invert with respect to the incircle of triangle ABC. The medial triangle of the intouch triangle is inverted...

shortlisted top 20 finalists, for one of the world's most prestigious Incircle Grant Schemes at the Abu Dhabi Middle East Film Festival 2007 Boot Polish...

(Minkowskian) triangle △ABC; if △ABC has an incircle s, then the six- (nine-) point circle S of △ABC touches its incircle s (Fig.173). In 2005 J. A. Scott used...

trigonometric identities. The largest circle contained in a triangle is called its incircle. It is known that: Δ = r 2 ⋅ ( cot ⁡ ∠ A 2 + cot ⁡ ∠ B 2 + cot ⁡ ∠ C 2...

Masayoshi and other from Jihei Morikawa were solved only recently. Equal incircles theorem Japanese theorem for concyclic polygons Japanese theorem for concyclic...

circles, are tangent to these three circles. They are separated by the incircle, one interior to it and one exterior. Descartes' theorem can be used to...

or three of the unspecified signs being negative. The equation of the incircle can be simplified to ± x cos ⁡ A 2 ± y cos ⁡ B 2 ± z cos ⁡ C 2 = 0 , {\displaystyle...

s={\tfrac {1}{2}}(a+b+c)} is the semiperimeter and r {\displaystyle r} the incircle radius. The identities can also be proven equivalent to the law of sines...

angle bisectors across the lines connecting pairs of centers of these incircles. The radius of each of the three Malfatti circles may be determined as...

holds for all Heronian triangles, but additionally the centers of the incircle and excircles are at lattice points. See also Integer triangle § Heronian...

all of whose sides are tangent to an incircle Bicentric quadrilateral – Convex, 4-sided shape with an incircle and a circumcircle Coxeter's loxodromic...

K_{T}=K{\frac {2r^{2}s}{abc}}} where K and r are the area and radius of the incircle, s is the semiperimeter of the original triangle, and a, b, c are the side...

The incircle of this dodecagram demonstrates that all the 14 VRPs are cocyclic, as alternatively shown by circle packings. The ratio of the incircle to...

theorem on the tangency of the nine-point circle of a triangle with its incircle and excircles Descartes' theorem on the radii of four pairwise tangent...

Carnot's theorem, which is used in a proof of the theorem above Equal incircles theorem Tangent lines to circles Johnson, Roger A., Advanced Euclidean...

primitive Heronian (non-Pythagorean) triangles with integer radii for the incircle and each excircle. A family of decomposable ones is given by a = 4 n 2...

given lines, there are four solutions to the general LLL problem (the incircle and excircles of the triangle formed by the three lines). Points and lines...

{\displaystyle l'} of △ A ′ B ′ C ′ {\displaystyle \triangle A'B'C'} (incircle of a colunar triangle) externally tangent to side A ′ B ′ . {\displaystyle...

evidently bounded by a constant R2 depending only on λ and ε. Now the incircle of a geodesic triangle has diameter less than δ where δ = 2 log 3; indeed...

proof of Feuerbach's theorem that the nine-point circle is tangent to the incircle and excircles of a triangle. Terquem's other contributions to mathematics...

orthocentric points. The common nine-point circle is tangent to all 16 incircles and excircles of the four triangles whose vertices form the orthocentric...

theorem Droz-Farny line theorem Encyclopedia of Triangle Centers Equal incircles theorem Equal parallelians point Equidissection Equilateral triangle Euler's...

other notable points, and has center on the nine-point circle. Inconics Incircle, the unique circle that is internally tangent to a triangle's three sides...

within each district, / As the beloved infant in his mother's bosom round incircled / With arms of love & pity & sweet compassion." (Milton. Book the second...