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...ian]] [[Victor Thébault]], individually known as Thébault's problem I, II, and III. ...m]] and a square version of the [[Napoleon's theorem]]. All three of these theorems are just a special case of [[Petr–Douglas–Neumann theorem]]. ...

...the radii of the green circles is equal to the sum of the radii of the red circles ...ter how one triangulates a cyclic ''quadrilateral'', the sum of inradii of triangles is constant. ...

...Geometry of Two Dimensions'', Forgotten Books, 2012 (orig. Deighton, Bell, and Co., 1866). http://www.forgottenbooks.com/search?q=Trilinear+coordinates&t The [[polar reciprocation|polar reciprocal]] and [[projective dual]] of this theorem give [[Pascal's theorem]]. ...

{{short description|Point where the incircle and nine-point circle of a triangle are tangent}} ...e [[nine-point circle]] is [[tangent circles|tangent]] to the [[incircle]] and [[excircles]] of a triangle. The incircle tangency is the Feuerbach point.] ...

...r's theorem''' states that the distance ''d'' between the [[circumcenter]] and [[incenter]] of a [[triangle]] is given by{{r|Johnson|Dunham}} ...adius and inradius respectively (the radii of the [[circumscribed circle]] and [[inscribed circle]] respectively). The theorem is named for [[Leonhard Eu ...

...y outside the triangle. In the diagram, ''DF'' is negative and both ''DG'' and ''DH'' are positive. [[Category:Theorems about triangles and circles]] ...

...mula]]'' can be thought as a special case of the Brahmagupta's formula for triangles. ...quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula. ...

{{Short description|Relationship between two figures of the same shape and size, or mirroring each other}} ...location and orientation, but leaves others unchanged, like [[distance]]s and [[angle]]s. The unchanged properties are called [[invariant (mathematics)|i ...

...|thumb|240px|right|The theorem states for any triangle {{math|∠ ''DAB''}} and {{math|∠ ''DAC''}} where AD is a bisector, then <math>|BD| : |CD| = |AB| ...t]] side {{mvar|{{overline|BC}}}} at a point {{mvar|D}} between {{mvar|B}} and {{mvar|C}}. The angle bisector theorem states that the ratio of the length ...

{{short description|Relates tangents of two angles of a triangle and the lengths of the opposing sides}} ...'γ''}} are respectively opposite the sides {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}.]] ...

...rcumcenter (black), altitudes and orthocenter (red), and nine-point circle and nine-point center (blue)]] ...(triangle)|altitudes]], and the points halfway between the [[orthocenter]] and each of the three vertices. The nine-point center is listed as point X(5) i ...

In [[mathematics]] and [[logic]], a '''direct proof''' is a way of showing the ...uantifier|''there exists'']]. Common proof rules used are [[modus ponens]] and [[universal instantiation]].<ref name="advanced-discrete-structure">C. Gupt ...

...e:ArcCapable.gif|thumb|class=skin-invert-image|For fixed points {{mvar|A}} and {{mvar|B}}, the set of points ''M'' in the plane, for which the angle {{mat ...agram at right. Choose two points on the circle, and call them {{mvar|V}} and {{mvar|A}}. Designate point {{mvar|B}} to be [[diametrically opposite]] po ...

{{Short description|On triangles inscribed in a circle with a diameter as an edge}} {{For|the theorem sometimes called Thales' theorem and pertaining to similar triangles|intercept theorem}} ...

...g|thumb|[[Farey sunburst]] of order 6, with 1 interior {{color|red|(red)}} and 96 boundary {{color|green|(green)}} points giving an area of {{nowrap|{{col ...book ''Mathematical Snapshots''.{{r|gs|steinhaus}} It has multiple proofs, and can be generalized to formulas for certain kinds of non-simple polygons. ...

{{Short description|Mathematical relation in spherical triangles}} ...<ref name=VNR/>) is a theorem relating the sides and angles of [[spherical triangles]], analogous to the ordinary [[law of cosines]] from plane [[trigonometry]] ...

...svg|thumb|360px|The two right triangles with leg and [[hypotenuse]] (7,13) and (13,17) have equal third sides of length <math>\sqrt{120}</math>. The squar ...uum problem''' is the problem of finding squares in arithmetic progression and their associated congrua. It can be formalized as a [[Diophantine equation] ...

{{about|the lengths of the sides of a triangle|his work on circles|Problem of Apollonius}} ...nn |first1=Alexander |last2=Wanner |first2=Gerhard |year=2012 |chapter=The Theorems of Apollonius–Pappus–Stewart |title=Geometry by Its History |publisher=Spri ...

...f a triangle intersect at the orthocenter, which for an [[Acute and obtuse triangles|acute triangle]] is inside the triangle.]] ...man|2001|loc=p. 118}}</ref> The orthocenter lies inside the triangle [[if and only if]] the triangle is [[acute triangle|acute]]. For a [[right triangle] ...

{{Short description|Branch of mathematics concerned with movement of shapes and sets}} ...cusing on [[group (mathematics)|groups]] of [[geometric transformation]]s, and properties that are [[invariant (mathematics)|invariant]] under them. It i ...