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Cevian
View on Wikipediafrom Wikipedia
In geometry, a cevian is a line segment which joins a vertex of a triangle to a point on the opposite side of the triangle.[1][2] Medians, symmedians, angle bisectors, altitudes are all special cases of cevians. The name cevian comes from the Italian mathematician Giovanni Ceva, who proved a theorem about cevians which also bears his name.[3]
Length
[edit]
Stewart's theorem
[edit]The length of a cevian can be determined by Stewart's theorem: in the diagram, the cevian length d is given by the formula
Less commonly, this is also represented (with some rearrangement) by the following mnemonic:
Median
[edit]If the cevian happens to be a median (thus bisecting a side), its length can be determined from the formula
or
since
Hence in this case
Angle bisector
[edit]If the cevian happens to be an angle bisector, its length obeys the formulas
and[5]
and
where the semiperimeter
The side of length a is divided in the proportion b : c.
Altitude
[edit]If the cevian happens to be an altitude and thus perpendicular to a side, its length obeys the formulas
and
where the semiperimeter
Ratio properties
[edit]
There are various properties of the ratios of lengths formed by three cevians all passing through the same arbitrary interior point:[6]: 177–188 Referring to the diagram at right,
The first property is known as Ceva's theorem. The last two properties are equivalent because summing the two equations gives the identity 1 + 1 + 1 = 3.
Splitter
[edit]A splitter of a triangle is a cevian that bisects the perimeter. The three splitters concur at the Nagel point of the triangle.
Area bisectors
[edit]Three of the area bisectors of a triangle are its medians, which connect the vertices to the opposite side midpoints. The medians all contain the centroid, which happens to lie at their common point of intersection, so that a uniform-density triangle would in principle balance on a razor supporting any of the medians.
Angle trisectors
[edit]If from each vertex of a triangle two cevians are drawn so as to trisect the angle (divide it into three equal angles), then the six cevians intersect in pairs to form an equilateral triangle, called the Morley triangle.
Area of inner triangle formed by cevians
[edit]Routh's theorem determines the ratio of the area of a given triangle to that of a triangle formed by the pairwise intersections of three cevians, one from each vertex.
See also
[edit]Notes
[edit]- ^ Coxeter, H. S. M.; Greitzer, S. L. (1967). Geometry Revisited. Washington, DC: Mathematical Association of America. p. 4. ISBN 0-883-85619-0.
- ^ Some authors exclude the other two sides of the triangle, see Eves (1963, p.77)
- ^ Lightner, James E. (1975). "A new look at the 'centers' of a triangle". The Mathematics Teacher. 68 (7): 612–615. JSTOR 27960289.
- ^ "Art of Problem Solving". artofproblemsolving.com. Retrieved 2018-10-22.
- ^ Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007 (orig. 1929), p. 70.
- ^ Alfred S. Posamentier and Charles T. Salkind, Challenging Problems in Geometry, Dover Publishing Co., second revised edition, 1996.
References
[edit]- Eves, Howard (1963), A Survey of Geometry (Vol. One), Allyn and Bacon
- Ross Honsberger (1995). Episodes in Nineteenth and Twentieth Century Euclidean Geometry, pages 13 and 137. Mathematical Association of America.
- Vladimir Karapetoff (1929). "Some properties of correlative vertex lines in a plane triangle." American Mathematical Monthly 36: 476–479.
- Indika Shameera Amarasinghe (2011). “A New Theorem on any Right-angled Cevian Triangle.” Journal of the World Federation of National Mathematics Competitions, Vol 24 (02), pp. 29–37.
Cevian
View on Grokipediafrom Grokipedia
In geometry, a cevian is a line segment that joins a vertex of a triangle to a point on the opposite side or its extension.[1] This concept generalizes several fundamental lines in a triangle, including altitudes (perpendicular from a vertex to the opposite side), medians (from a vertex to the midpoint of the opposite side), and angle bisectors (dividing the angle at a vertex equally).[1] Cevians play a central role in triangle geometry, enabling the study of concurrency, ratios, and area relations within triangles.[2]
The term "cevian" derives from the name of the Italian mathematician and engineer Giovanni Ceva (1647–1734), who formalized key properties of such lines in his 1678 work De lineis rectis.[2] Although the geometric ideas predate Ceva—such as the proof by the 11th-century mathematician Yusuf al-Mu’taman ibn Hūd in his Kitab al-Istikmal—the modern usage honors Ceva's contributions to concurrency conditions.[3] Cevians extend beyond acute triangles to obtuse and even degenerate cases, where the intersection point may lie outside the triangle.[1]
A cornerstone theorem associated with cevians is Ceva's theorem, which provides a necessary and sufficient condition for three cevians—one from each vertex—to be concurrent (intersect at a single point).[3] Specifically, for triangle with cevians , , and (where , , and lie on the opposite sides , , and respectively), the cevians concur if and only if , using directed segment lengths.[2] This result, provable via area ratios or trigonometric identities, has profound implications for classifying triangle centers and solving concurrency problems.[3] Extensions of Ceva's theorem appear in higher-dimensional geometry and projective spaces, underscoring its enduring influence.[4]
where denotes the area of the triangle.[42] This formula derives directly from the basic area expression for a triangle, , which equates the area to half the product of the base and corresponding height; solving for the height yields the altitude length.[42] When the side lengths , , and are known but angles are not, the area may be computed using Heron's formula,
where is the semiperimeter, leading to the explicit altitude expression
[43][42] In specific triangle types, the altitude length simplifies further. For an equilateral triangle with side length , the altitude is
reflecting the symmetry and 60° angles.[44] In a right triangle with the right angle at vertex and legs of lengths (adjacent to ) and (adjacent to ), the altitude from to leg equals the leg length , and similarly, the altitude from to leg equals .[45] Altitudes relate to the orthocenter , the concurrency point of the three altitudes, which divides each altitude into two segments. The product of the lengths of these segments is the same for all three altitudes and holds across acute, right, and obtuse triangles; in the obtuse case, where lies outside the triangle, the division occurs externally for two altitudes, but the product property persists using directed or absolute lengths.[46]
Definition and History
Definition
In geometry, a cevian is a line segment that joins a vertex of a triangle to a point on the opposite side or its extension.[1] This construction is fundamental to the study of triangles, as it connects a vertex directly to the base opposite it, allowing for the subdivision of the triangle into smaller triangular regions. Standard notation for a cevian in triangle designates the cevian from vertex to point on side as , where , , and .[5] If lies between and , the cevian is internal and divides side into two positive segments summing to ; if is outside the segment , such as beyond or , the cevian is external, with one of or being negative in directed segment conventions to indicate the extension. Special cases of cevians include medians, where is the midpoint of ; altitudes, where is perpendicular to ; angle bisectors, where divides in the ratio of the adjacent sides; and symmedians, which are the isogonal conjugates of the medians.[1] These examples illustrate the versatility of cevians in capturing symmetric or perpendicular relationships within the triangle. Cevians serve a key role in triangle geometry by linking vertices to bases, enabling analyses of area division, balance points, and concurrency conditions for multiple cevians.[6]History
The study of cevians traces its roots to ancient Greek geometry, where special cases such as medians, altitudes, and angle bisectors were well understood. Medians, connecting a vertex to the midpoint of the opposite side, were implicitly recognized in Euclid's Elements (c. 300 BCE), particularly in discussions of triangle properties and bisections in Books I and IV.[7] Altitudes, perpendicular lines from vertices to opposite sides, were explored by Euclid in propositions on right triangles and area calculations, such as Proposition 12 in Book II. The angle bisector theorem, which states that the bisector divides the opposite side in the ratio of the adjacent sides, appears as Proposition 3 in Book VI of Euclid's Elements (c. 300 BCE).[8] A key precursor to cevian concurrency is Menelaus's theorem (c. 100 CE), which addresses transversals intersecting the sides of a triangle, laying groundwork for later concurrency conditions. The theorem is attributed to Menelaus of Alexandria, though his surviving work Sphaerica focuses on spherical geometry.[9] The general concept of cevian concurrency emerged in Islamic mathematics during the 11th century. Yusuf al-Mu'taman ibn Hud, a ruler and scholar in Zaragoza, provided the first known proof of what is now called Ceva's theorem in his Kitab al-Istikmal, demonstrating conditions under which three cevians from vertices meet at a point.[3] This result was independently rediscovered by Italian mathematician Giovanni Ceva (1647–1734), who formalized it in his 1678 treatise De lineis rectis, focusing on ratios along triangle sides without trigonometric methods.[10] In the 18th and 19th centuries, developments shifted toward cevian lengths and areas. Scottish mathematician Matthew Stewart published a theorem in 1746 relating cevian lengths to side lengths in Some General Theorems of Considerable Use in the Higher Parts of Geometry, though it gained wider recognition later.[5] British mathematician Edward John Routh advanced area-related properties in 1878, deriving formulas for the area of the triangle formed by three cevians in solutions to Cambridge problems.[11] The term "cevian" itself, derived from Giovanni Ceva's name, was coined in French as cévienne in 1888 by A. Poulain and entered English usage in the early 20th century to unify these generalized lines in triangle geometry.[12]Special Types of Cevians
Medians
A median of a triangle is a special type of cevian that joins a vertex to the midpoint of the opposite side.[13] In any triangle, there are exactly three medians, one from each vertex. These lines play a fundamental role in triangle geometry due to their concurrency and balancing properties. The three medians intersect at a single point known as the centroid, which divides each median into two segments in the ratio 2:1, with the longer portion adjacent to the vertex.[14] This division point serves as the geometric center of the triangle. Furthermore, the medians partition the triangle into six smaller triangles of equal area, highlighting their role in equitable spatial division.[15] The centroid also functions as the balance point, or center of mass, for a triangle assuming uniform density, making it the point through which the triangle's weight appears to act when suspended.[16] As area-bisecting cevians, medians uniquely ensure that each divides the original triangle into two regions of equal area, a property intrinsic to their connection to midpoints.[17]Altitudes
An altitude of a triangle is a cevian from a vertex that is perpendicular to the line containing the opposite side, with the foot of the altitude being the point of intersection on that side or its extension.[18] In obtuse triangles, the foot may lie outside the segment of the opposite side.[19] The three altitudes of a triangle are concurrent at a point known as the orthocenter.[19] The position of the orthocenter relative to the triangle depends on the type of triangle: it lies inside the triangle for acute triangles, at the vertex of the right angle for right triangles, and outside the triangle for obtuse triangles.[20] The feet of the altitudes form the orthic triangle, which is the pedal triangle of the orthocenter.[21] This orthic triangle plays a key role in triangle geometry, and the orthocenter itself is a pivotal point on the Euler line, which connects the orthocenter, the centroid, and the circumcenter of the triangle.[22] Altitudes uniquely determine the area of the triangle through the conceptual relation that the area equals half the product of any side (as base) and the corresponding altitude (as height).[23]Angle Bisectors
In a triangle, an angle bisector is a cevian that originates from a vertex and divides the angle at that vertex into two equal angles, extending to intersect the opposite side.[24] This cevian plays a fundamental role in triangle geometry by facilitating the identification of key internal points and supporting theorems about side divisions. The three internal angle bisectors of a triangle are concurrent, intersecting at a single point known as the incenter, which serves as the center of the triangle's incircle—the largest circle tangent to all three sides internally.[25] The incenter is uniquely characterized by its trilinear coordinates of 1:1:1 relative to the triangle's vertices, and it lies at equal perpendicular distances from all sides, equal to the inradius.[25] The points of tangency of the incircle with the sides are determined by the angle bisectors, as these cevians guide the equal division of angles and thus the symmetric placement of the incircle. A central property of the angle bisector cevian is encapsulated in the angle bisector theorem, which states that if the bisector from vertex in meets side at point , then it divides in the ratio of the adjacent sides: .[8] This ratio property underscores the bisector's role in apportioning the opposite side proportionally to the flanking sides, providing a geometric basis for coordinate systems like tangential coordinates where the incenter acts as a reference point. In addition to internal bisectors, external angle bisectors—those dividing the exterior angles at each vertex—also form cevians that extend beyond the triangle. The three external angle bisectors concur at the excenters, with each excenter serving as the center of an excircle tangent to one side of the triangle and the extensions of the other two.[26] There are three such excenters, forming an orthocentric system with the incenter, and they highlight the duality between internal and external angular divisions in triangle geometry.Symmedians
Symmedians are cevians in a triangle that are the reflections of the medians over the respective angle bisectors. For a triangle with sides , , and , the symmedian from vertex intersects side at point such that .[27] This division ratio distinguishes symmedians from other cevians like medians (which divide in the ratio 1:1) and angle bisectors (which divide in the ratio c:b internally).[28] The three symmedians of a triangle are concurrent at a point known as the symmedian point or Lemoine point, with trilinear coordinates and barycentric coordinates .[29] This concurrency was established in the 19th century, building on earlier observations of the point minimizing the sum of squared distances to the sides.[30] Symmedians are isogonal conjugates of the medians, meaning each symmedian is the reflection of the corresponding median in the angle bisector at its vertex.[27] Key properties include the fact that points on a symmedian maintain proportional distances to the adjacent sides: for a point on the symmedian from , the ratios of distances from to sides and equal .[28] Symmedians also relate to antiparallels, as the symmedian from serves as the median in triangles formed by antiparallels to side .[29] In right-angled triangles, the symmedian from the right-angled vertex to the hypotenuse coincides with the altitude to the hypotenuse.[27] The concept emerged in the early 19th century, with initial characterizations in 1803–1804 by figures like James Ivory, who described the symmedian point's minimizing property, though the lines themselves were formalized later by Émile Lemoine in 1873 as "antiparallel medians," with the term "symmedian" coined by Maurice d’Ocagne in 1883.[30]Lengths of Cevians
Stewart's Theorem
Stewart's theorem provides a general relation for the length of a cevian in a triangle. Consider triangle with sides , , and . Let be a point on side such that and , where . The cevian has length . Then, Stewart's theorem states that [5][31] The theorem is named after the Scottish mathematician Matthew Stewart, who published it in 1746 as one of several general geometric propositions in his work Some General Theorems of Considerable Use in the Higher Parts of Mathematics.[32][33] This publication contributed to Stewart's reputation in geometry during the 18th century.[32] A standard derivation uses the law of cosines applied to the sub-triangles and . Let be the angle at in , so the angle at in is and . Applying the law of cosines in : yielding In : Substituting the expression for : Multiplying through by : Rearranging terms: [31][5] Stewart's theorem serves as a fundamental tool for computing cevian lengths given the side lengths and division segments of the base, applicable to any cevian in a triangle.[5] Alternative proofs exist using vector methods or coordinate geometry, placing the triangle in the plane and applying the distance formula.[31] For instance, when is the midpoint of (so ), the theorem simplifies to verify the median length formula: confirming consistency with known results for medians.[5]Length of the Median
The length of the median from vertex to the midpoint of side (of length ), where the adjacent sides are and , is derived by specializing Stewart's theorem to the case where the cevian divides the opposite side into equal segments . Substituting these values into the theorem produces the formula [34][35] This expression, equivalent to , is a direct consequence of the theorem's general relation for cevians and provides the precise length in terms of the triangle's side lengths. Apollonius's theorem formalizes this relation as a generalization for medians, stating that in any triangle, the sum of the squares of two sides equals twice the square of the median to the third side plus twice the square of half the third side: . Rearranging yields the same median length formula, emphasizing its role as a foundational metric relation independent of angles.[36] A key property arises from applying the formula to all three medians , , and : their squares sum to three-quarters of the sum of the squares of the sides, This is obtained by adding the individual squared median formulas and simplifying. The relation extends to vector formulations involving the centroid , where the sum of squared distances from to the vertices is ; since each such distance is of the corresponding median length, it connects the medians to the centroid's position vector as the average of the vertices' position vectors.[37][38] For example, in an equilateral triangle with side length , all medians coincide with the altitudes, yielding . Substituting into the sum-of-squares property confirms , or .[39]Length of the Angle Bisector
The length of the angle bisector from vertex A to the opposite side BC in triangle ABC, with sides , , , and angle at A, is given by This trigonometric form arises from applying the law of cosines in the sub-triangles formed by the bisector and using the half-angle property.[40] An equivalent algebraic expression, independent of the angle, is which can also be written as This form follows directly from the angle bisector theorem, which states that the bisector divides BC into segments BD and DC with (or where , ), so , . Substituting these ratios into Stewart's theorem for the cevian length yields the result.[41] Alternatively, the formula can be derived using area considerations or triangle similarity without invoking Stewart's theorem explicitly. For instance, the areas of triangles ABD and ACD are equal when expressed in terms of the bisected angle halves, leading to the same squared length after simplification.[40] In an isosceles triangle where , the angle bisector from A coincides with the median and altitude due to symmetry, simplifying the formula to .[40] For an equilateral triangle with side length , the angle bisector length simplifies to the altitude , as , , and , so .[40]Length of the Altitude
The length of the altitude from vertex to side of length in a triangle is given bywhere denotes the area of the triangle.[42] This formula derives directly from the basic area expression for a triangle, , which equates the area to half the product of the base and corresponding height; solving for the height yields the altitude length.[42] When the side lengths , , and are known but angles are not, the area may be computed using Heron's formula,
where is the semiperimeter, leading to the explicit altitude expression
[43][42] In specific triangle types, the altitude length simplifies further. For an equilateral triangle with side length , the altitude is
reflecting the symmetry and 60° angles.[44] In a right triangle with the right angle at vertex and legs of lengths (adjacent to ) and (adjacent to ), the altitude from to leg equals the leg length , and similarly, the altitude from to leg equals .[45] Altitudes relate to the orthocenter , the concurrency point of the three altitudes, which divides each altitude into two segments. The product of the lengths of these segments is the same for all three altitudes and holds across acute, right, and obtuse triangles; in the obtuse case, where lies outside the triangle, the division occurs externally for two altitudes, but the product property persists using directed or absolute lengths.[46]