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...gle's vertices together with <math>P</math> form a concyclic quadrilateral and hence Ptolemy's theorem yields: *Doug French: ''Teaching and Learning Geometry''. Bloomsbury Publishing, 2004, {{ISBN|9780826434173}} , ...

{{short description|On rays from a point to a line, with equal inscribed circles between adjacent rays}} [[Image:Incircles.JPG|thumb|360px|If the blue circles are equal, the green circles are also equal.]] ...

{{short description|Concurrency of lines connecting to certain circles associated with an arbitrary triangle}} ...ms that the three [[straight line]]s <math>AO_a</math>, <math>BO_b</math>, and <math>CO_c</math> are concurrent.<ref name=wolframKosnita/> This result was ...

...more, let ''E'' and ''F'' the midpoints of its diagonals ''AC'' and ''BD'' and ''P'' be the center of its incircle. Given such a configuration the point P ...bus. In this case, both midpoints and the center of the incircle coincide, and by definition, no Newton line exists. ...

...t description|Concerns 3 circles through triples of points on the vertices and sides of a triangle}} ...ssing through the vertices of a triangle ''ABC'' and points ''A´'', ''B´'' and ''C´'' on the adjacent sides of the triangle intersecting at a common point ...

...n of [[line segment]]s created by two intersecting [[Secant line|secant]]s and the associated [[circle]]. ...ar|BC}} that [[Line–line intersection|intersect each other]] at {{mvar|P}} and for which {{math|''A'', ''B'', ''C'', ''D''}} all lie on the same circle, t ...

More precisely, for two chords {{mvar|AC}} and {{mvar|BD}} intersecting in a point {{mvar|S}} the following equation holds ...[[diagonal]]s of a [[quadrilateral]] {{mvar|ABCD}} intersect in {{mvar|S}} and fulfill the equation above, then it is a [[cyclic quadrilateral]]. ...

{{short description|About a common point of certain circles defined by an arbitrary triangle}} ...,C^*</math>, respectively. The theorem says that these three '''Musselman circles''' meet in a point <math>M</math>, that is the [[circle inversion|inverse]] ...

...le|incircle]] (dashed gray), and the centre of both circles (white); solid and dashed [[Line segment|line segments]] of the same colour are equal in lengt ...om.fau.edu/FG2013volume13/FG2013volume13.pdf#page=195}}</ref> The theorem and circle are named after mathematician [[John Horton Conway]]. ...

...> intersect in <math>D</math> <br/> <math>AP'_a</math>, <math>AP'_b</math> and <math>AP'_c</math> intersect in <math>D'</math>]] ...a property of the [[cevian]]s of a triangle intersecting in a common point and is named after the German mathematician [[Karl Gustav Reuschle]] (1812–1875 ...

...rt description|Geometry theorem relating line segments created by a secant and tangent line}} ...ibes the relation of [[line segment]]s created by a [[Secant line|secant]] and a [[tangent]] line with the associated [[circle]]. ...

{{short description|Area of a triangle from its sides and vertex distances to any line tangent to its incircle}} ...ry]] for the [[area]] of a [[triangle]], as a function of its side lengths and the perpendicular distances of its vertices from an arbitrary line tangent ...

...rt description|A statement about properties of inscribed and circumscribed circles}} ...]] that the [[line segment]] between the [[incircle and excircles|incenter and any excenter]] of a triangle, or between two excenters, is the [[diameter]] ...

...|publisher=John Wiley & Sons|year=1969|location = New York, London, Sydney and Toronto|pages=72–75}}</ref> ...> Expressing the rotation by a [[linear transformation]] <math>T(x)</math> and the dilation as multiplying by a scale factor <math>d</math>, a point <math ...

In [[mathematics]], '''Ky Fan's lemma (KFL)''' is a combinatorial lemma about labellings of triangulations. It is a generalization of [[Tucker's lemma]]. ...tary edge'' of ''L'' if the labels of its two endpoints have the same size and opposite signs, e.g. {−2, +2}. ...

...diagonals of the quadrilateral remain diagonals of the complete quadrangle and the Newton line of the quadrilateral is the Newton–Gauss line of the comple ...ttps://www.researchgate.net/publication/266061112|title=Gauss–Newton Lines and Eleven Point Conics|last=Alperin|first=Roger C.|author-link=Roger C. Alperi ...

...terior angle theorem" as the ''strong form'' of the exterior angle theorem and "Euclid's exterior angle theorem" as the ''weak form''.<ref>{{harvnb|Wylie| ...ding one of the sides of the triangle; the angle between the extended side and the other side is the exterior angle. In the picture, angle ''∠ACD'' is an ...

...name="Davis">{{cite journal |last1=Philip J. Davis |title=The Rise, Fall, and Possible Transfiguration of Triangle Geometry: A Mini-history |journal=The ...and twentieth centuries: the new triangle geometry in textbooks in Europe and USA (1888–1952) |contribution-url=https://skemman.is/bitstream/1946/26925/1 ...

{{short description|Characterizes spherical triangles with fixed base and area}} ...l circle (dashed green) passing through the points antipodal to {{mvar|A}} and {{mvar|B}}.]] ...

{{Short description|Property of all triangles on a Euclidean plane}} {{About|the trigonometric identity|the cosine law of optics|Lambert's cosine law}} ...