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Ceva's theorem

Ceva's theorem is a fundamental result in Euclidean geometry concerning the concurrency of cevians in a triangle. For a triangle \triangle ABC with cevians AD, BE, and CF (where D lies on side BC, E on CA, and F on AB), the cevians are concurrent if and only if

\frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB} = 1.

This condition holds using directed segment lengths, allowing for extensions to cases where cevians are parallel or points lie outside the sides.^[1]^[2]^[3] The theorem was first published in 1678 by Italian mathematician and engineer Giovanni Ceva (1647–1734) in his work De lineis rectis, where it appeared as a lemma in a broader study of triangle geometry, though Ceva himself provided no proof.^[1]^[3] Although first published by Ceva, the theorem was proven centuries earlier by the Islamic mathematician Yusuf al-Mu'taman ibn Hud (d. 1085).^[4] A trigonometric form of Ceva's theorem, useful in contexts involving angles, states that the cevians are concurrent if and only if

\frac{\sin \angle BAD}{\sin \angle CAD} \cdot \frac{\sin \angle CBE}{\sin \angle ABE} \cdot \frac{\sin \angle ACF}{\sin \angle BCF} = 1.

This variant follows from the law of sines applied to the sub-triangles formed by the cevians.^[5]^[6] Proofs of the theorem vary, including area-based methods using ratios of triangle areas sharing the same height, barycentric coordinate approaches that assign masses to vertices for balance at the intersection point, and vector or projective geometry techniques that generalize to higher dimensions.^[2]^[3] Ceva's theorem has broad applications in triangle geometry, establishing the concurrency of special cevians such as medians (at the centroid), altitudes (at the orthocenter), and angle bisectors (at the incenter).^[1] It also connects to related results like Menelaus's theorem on transversal lines and extends to spherical and hyperbolic geometries, as well as n-dimensional simplices via generalizations involving products of ratios along faces.^[2] The theorem's elegance and utility have made it a cornerstone for studying triangle centers and cevian nests in classical and modern geometry.^[3]

Statement and Notation

Classical Formulation

In a triangle ABC, a cevian is a line segment joining a vertex to a point on the opposite side. Specifically, the cevians AD, BE, and CF connect vertices A, B, and C to points D, E, and F on the opposite sides BC, CA, and AB, respectively, where D, E, and F divide the sides into interior segments.^[7]^[1] Ceva's theorem states that these cevians are concurrent—that is, they intersect at a single interior point—if and only if the product of the ratios of the lengths of the segments on each side equals unity. Using the notation where segment lengths are denoted by overlines (e.g., \overline{BD}), the condition is

\frac{\overline{BD}}{\overline{DC}} \cdot \frac{\overline{CE}}{\overline{EA}} \cdot \frac{\overline{AF}}{\overline{FB}} = 1,

with all ratios being unsigned positive real numbers corresponding to the lengths of the interior divisions.^[1]^[8] This formulation assumes that points D, E, and F lie strictly between the vertices on each side, ensuring the cevians are internal and the concurrency point is inside the triangle.^[9] The theorem is typically illustrated by a diagram of triangle ABC with cevians AD, BE, and CF drawn from each vertex to the labeled points on the opposite sides, all three lines meeting at one common interior point.^[1]

Signed Ratios and Converse

In the signed version of Ceva's theorem, the ratios of the segments are interpreted as directed quantities, allowing the theorem to apply to configurations where the intersection points on the sides may lie outside the triangle segments./04:_Elementary_Euclidean_Geometry/4.03:_Theorems_of_Ceva_and_Menelaus) Signed ratios are positive when a point divides a side internally (between the vertices) and negative when it divides externally (beyond one of the vertices), with the sign determined by the chosen orientation of the line.^[2] This directed approach uses vector notation to represent the segments, where \overrightarrow{BD} denotes the directed segment from B to D along line BC, and similarly for the others.^[2] The theorem states that the cevians AD, BE, and CF are concurrent if and only if the product of the signed ratios equals 1:

\frac{\overrightarrow{BD}}{\overrightarrow{DC}} \cdot \frac{\overrightarrow{CE}}{\overrightarrow{EA}} \cdot \frac{\overrightarrow{AF}}{\overrightarrow{FB}} = 1.

/04:_Elementary_Euclidean_Geometry/4.03:_Theorems_of_Ceva_and_Menelaus) Here, the vector notation ensures consistency in direction; for instance, if the positive direction on BC is from B to C, then an external division beyond C would make \overrightarrow{DC} negative relative to \overrightarrow{BD}, adjusting the ratio accordingly to account for the cevian extension.^[2] This formulation generalizes the classical case by handling all positions of D, E, and F on the lines containing the sides, including points at infinity where cevians are parallel (a degenerate concurrent case)./04:_Elementary_Euclidean_Geometry/4.03:_Theorems_of_Ceva_and_Menelaus) The converse is inherently biconditional in this signed version: if the product of the signed ratios equals 1, then the cevians AD, BE, and CF are concurrent (or all parallel).^[2] This bidirectional statement provides a complete characterization of concurrency via the algebraic condition, proven by assuming two cevians intersect and verifying the third passes through the same point using the ratio equality./04:_Elementary_Euclidean_Geometry/4.03:_Theorems_of_Ceva_and_Menelaus) For example, consider a configuration where point D lies on the extension of BC beyond C (exterior to segment BC), while E and F are interior to CA and AB, respectively. If the directions are oriented from B to C, C to A, and A to B, then \overrightarrow{BD} remains positive but \overrightarrow{DC} becomes negative due to the reversal across C, yielding a negative ratio \frac{\overrightarrow{BD}}{\overrightarrow{DC}}.^[2] To achieve concurrency at a point inside the triangle, the positions of E and F can be adjusted such that the positive ratios \frac{\overrightarrow{CE}}{\overrightarrow{EA}} and \frac{\overrightarrow{AF}}{\overrightarrow{FB}} compensate, making the overall product equal to 1. This demonstrates how negative signs enable the theorem to capture concurrency even when one cevian originates from an external division./04:_Elementary_Euclidean_Geometry/4.03:_Theorems_of_Ceva_and_Menelaus)

Historical Development

Early Islamic Contributions

During the Islamic Golden Age, particularly in the region of al-Andalus, scholars advanced geometric knowledge through systematic studies of proportions and concurrency in triangles, building on Greek foundations while introducing innovative proofs. Yusuf al-Mu'taman ibn Hud (died 1085), who ruled Zaragoza from 1081 to 1085, exemplified this progress in his comprehensive treatise Kitab al-Istikmal (Book of Perfection), an encyclopedic work on arithmetic and geometry that included the first known proof of the theorem now associated with Ceva.^[10] As both a patron of learning and a practicing mathematician, Ibn Hud integrated classical sources like Euclid's Elements with original analyses, reflecting the vibrant intellectual environment of 11th-century Islamic Spain.^[11] Ibn Hud's contribution focused on concurrent lines drawn from the vertices of a triangle to points on the opposite sides, demonstrating that these cevians meet at a single point if and only if the product of certain segment ratios equals unity—a condition central to understanding triangle concurrency. His proof employed area-based methods, aligning with broader Islamic geometric traditions that emphasized visual and proportional reasoning to extend Hellenistic results. This formulation appeared within a larger compilation of geometric propositions, showcasing Ibn Hud's synthesis of inherited knowledge and novel insights into plane geometry.^[10] The Kitab al-Istikmal circulated in manuscript form during the medieval period but largely escaped notice in Europe, remaining confined to Islamic scholarly networks until its contents were systematically reconstructed and analyzed. In 1985, historian of mathematics Jan P. Hogendijk identified and examined surviving fragments, primarily through a 14th-century redaction by Ibn Sartaq, revealing Ibn Hud's theorem as a precursor to later European discoveries.^[11] This rediscovery highlighted the theorem's independent origins in Islamic mathematics, predating Giovanni Ceva's 1678 publication by over six centuries.

Ceva's Publication and Rediscovery

Giovanni Ceva (1647–1734), an Italian mathematician and engineer, first published the theorem in his 1678 work De lineis rectis se invicem secantibus statica constructio (On the Static Construction of Straight Lines Intersecting Each Other), where it appeared as a lemma supporting investigations into statics and the equilibrium of forces. Ceva presented the result without a formal geometric proof, instead deriving it through mechanical considerations related to centers of gravity and concurrent lines in triangles, reflecting the era's integration of geometry with practical engineering problems.^[10] In the broader European mathematical landscape of the late 17th century, Ceva's formulation addressed issues of concurrent forces and the division of triangles by transversals, though Ceva showed no awareness of earlier Islamic mathematical contributions that had already established the result. This European rediscovery occurred amid a revival of synthetic geometry, where such theorems facilitated analyses in optics, mechanics, and architecture, yet the work's limited circulation meant its influence remained modest until the 19th century. Despite predating proofs in Islamic scholarship—such as those from the 11th century—the theorem bears Ceva's name eponymously due to his role in introducing it to Western Europe, a convention solidified in subsequent geometric literature.^[10] Full historical attribution, recognizing earlier origins, emerged through 20th-century scholarship, notably the 1985 rediscovery and analysis of Ibn Hud's manuscript demonstrating the theorem's prior establishment.^[11] Ceva's specific presentation employed unsigned ratios of line segments for cevians intersecting at interior points of the triangle, a choice that simplified applications in statics but later prompted extensions to signed ratios in more general cases; this formulation influenced 18th- and 19th-century texts on Euclidean geometry, including those by Euler and Poncelet.

Geometric Interpretation

Cevians and Concurrency

In geometry, a cevian is defined as a line segment that connects a vertex of a triangle to a point on the opposite side, which may include extensions beyond the side if necessary.^[12] This construction allows for the exploration of internal divisions within the triangle, where the endpoint on the opposite side can vary, influencing the geometric properties of the figure.^[13] Concurrency refers to the geometric property in which three cevians—one originating from each vertex of the triangle—intersect at a single common point, either inside the triangle or at an exterior location depending on the configuration.^[14] This intersection point, often denoted as the concurrency point, unifies the cevians and is central to theorems governing triangle concurrency.^[12] Consider a triangle ABC with cevians AD, BE, and CF, where D lies on side BC, E on AC, and F on AB; these cevians meet at a point O if they are concurrent, and Ceva's theorem provides the ratio product condition that characterizes this concurrence.^[15] The setup highlights how the positions of D, E, and F determine whether the cevians converge, offering a framework for analyzing intersection behaviors in triangular configurations.^[2] Cevian properties extend to special cases such as medians, which connect vertices to midpoints; altitudes, perpendicular to opposite sides; and angle bisectors, dividing angles equally; each set exhibits concurrency under particular ratio conditions derived from Ceva's theorem.^[16] These variants illustrate the versatility of cevians in revealing symmetric or perpendicular intersections within the triangle.

Illustrative Examples

One classic illustration of Ceva's theorem involves the medians of a triangle. In triangle ABC, a median from vertex A connects A to the midpoint D of side BC, so \frac{BD}{DC} = 1; similarly, the median from B to midpoint E of AC gives \frac{CE}{EA} = 1, and from C to midpoint F of AB gives \frac{AF}{FB} = 1. The product of these ratios is (1)(1)(1) = 1, satisfying Ceva's condition and confirming that the medians are concurrent at the centroid.^[2] Another example is the altitudes in an acute triangle ABC. The altitude from A meets BC at D, from B meets AC at E, and from C meets AB at F. The ratios \frac{BD}{DC}, \frac{CE}{EA}, and \frac{AF}{FB} depend on the side lengths of the triangle—for instance, in a specific acute triangle with sides a, b, c opposite vertices A, B, C, these ratios can be expressed using trigonometric identities or areas, but their product equals 1 regardless, ensuring concurrency at the orthocenter.^[17] Ceva's theorem also applies to the angle bisectors of triangle ABC. The angle bisector from A intersects BC at D, dividing it in the ratio of the adjacent sides \frac{BD}{DC} = \frac{AB}{AC} by the angle bisector theorem; similarly, \frac{CE}{EA} = \frac{BC}{BA} and \frac{AF}{FB} = \frac{CA}{CB}. The product simplifies to \left(\frac{AB}{AC}\right)\left(\frac{BC}{BA}\right)\left(\frac{CA}{CB}\right) = 1, verifying that the bisectors concur at the incenter. To see the necessity of the product equaling 1, consider a counterexample where the condition fails. Suppose points D, E, F on sides BC, CA, AB of triangle ABC are chosen such that \frac{BD}{DC} = 2, \frac{CE}{EA} = 3, and \frac{AF}{FB} = 1, yielding a product of $6 \neq 1. By the converse of Ceva's theorem, the cevians AD, BE, CF are not concurrent, as the lines intersect pairwise but not at a single point.^[2]

Proofs

Area-Based Proof

Consider triangle ABC with cevians AD, BE, and CF intersecting at an interior point O, where D lies on side BC, E on CA, and F on AB. Denote the area of triangle XYZ by [XYZ]. This proof assumes O is interior to ABC, ensuring all relevant areas are positive. Triangles ABD and ACD share the altitude from A to line BC, so their areas are proportional to bases BD and DC:

\frac{BD}{DC} = \frac{[ABD]}{[ACD]}.

Analogously,

\frac{CE}{EA} = \frac{[BCE]}{[BAE]}, \quad \frac{AF}{FB} = \frac{[CAF]}{[CBF]}.

Since O lies on cevian AD, triangles AOB and ABD share the altitude from B to line AD, yielding

\frac{[AOB]}{[ABD]} = \frac{AO}{AD}.

Likewise, \frac{[AOC]}{[ACD]} = \frac{AO}{AD}. Thus,

\frac{[AOB]}{[AOC]} = \frac{[AOB]}{[ABD]} \cdot \frac{[ACD]}{[AOC]} = \frac{[ABD]}{[ACD]} = \frac{BD}{DC}.

For cevian BE, triangles BOC and BCE share the altitude from C to BE, so \frac{[BOC]}{[BCE]} = \frac{BO}{BE}. Similarly, \frac{[BOA]}{[BAE]} = \frac{BO}{BE}. Therefore,

\frac{[BOC]}{[BOA]} = \frac{[BCE]}{[BAE]} = \frac{CE}{EA}.

For cevian CF, triangles COA and CAF share the altitude from A to CF, so \frac{[COA]}{[CAF]} = \frac{CO}{CF}. Similarly, \frac{[COB]}{[CBF]} = \frac{CO}{CF}. Therefore,

\frac{[COA]}{[COB]} = \frac{[CAF]}{[CBF]} = \frac{AF}{FB}.

Now multiply the three area ratios:

\frac{[AOB]}{[AOC]} \cdot \frac{[BOC]}{[BOA]} \cdot \frac{[COA]}{[COB]} = \frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB}.

The left side simplifies as follows (noting [BOA] = [AOB], [COA] = [AOC], and [COB] = [BOC]):

\frac{[AOB]}{[AOC]} \cdot \frac{[BOC]}{[AOB]} \cdot \frac{[AOC]}{[BOC]} = 1.

Hence,

\frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB} = 1,

establishing Ceva's theorem under the concurrency assumption. The converse follows similarly by reversing the area relations.

Barycentric Coordinate Proof

Barycentric coordinates offer a coordinate-free method well-suited to Ceva's theorem, as they directly encode the ratios along the triangle's sides through area proportions. In the plane of triangle ABC, a point P has barycentric coordinates (a : b : c) with respect to vertices A, B, C, normalized such that a + b + c = 1. These coordinates represent the relative areas of the sub-triangles formed by P and the sides: a = [\triangle PBC]/[\triangle ABC], b = [\triangle PCA]/[\triangle ABC], and c = [\triangle PAB]/[\triangle ABC], where [ \cdot ] denotes signed area.^[18] For the points defining the cevians in Ceva's theorem, consider D on side BC, E on CA, and F on AB. The point D has coordinates (0 : b_D : c_D) with b_D + c_D = 1, where b_D / c_D = DC / BD, reflecting the area ratio [\triangle DCA] : [\triangle DAB] = DC : BD. Thus, b_D = DC / (BD + DC) and c_D = BD / (BD + DC). Similarly, E has coordinates (a_E : 0 : c_E) with a_E + c_E = 1 and a_E / c_E = CE / EA, so a_E = CE / (CE + EA) and c_E = EA / (CE + EA). For F, the coordinates are (a_F : b_F : 0) with a_F + b_F = 1 and a_F / b_F = FB / AF, yielding a_F = FB / (AF + FB) and b_F = AF / (AF + FB).^[19] The cevians AD, BE, and CF are concurrent if and only if there exists a point O with consistent barycentric coordinates when parameterized along each cevian. Parameterize the line AD as O = t A + (1 - t) D for t \in (0,1), giving coordinates (t : (1-t) b_D : (1-t) c_D). Along BE, O = s B + (1 - s) E = ((1-s) a_E : s : (1-s) c_E). Along CF, O = u C + (1 - u) F = ((1-u) a_F : (1-u) b_F : u). Setting these equal yields a system where the ratios must match.^[18] Equating the coordinates from AD and BE (and similarly for the third), the condition simplifies through the ratios. Specifically, the concurrency holds if and only if \frac{b_D}{c_D} \cdot \frac{c_E}{a_E} \cdot \frac{a_F}{b_F} = 1. Substituting the expressions, this becomes \frac{DC}{BD} \cdot \frac{EA}{CE} \cdot \frac{FB}{AF} = 1, which is equivalent to the classical form \frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB} = 1. This derivation leverages the affine invariance of barycentric coordinates, providing a general proof without relying on specific metrics.^[19]

Trigonometric Proof

The trigonometric proof of Ceva's theorem derives the condition for concurrency of cevians AD, BE, and CF in triangle ABC by applying the law of sines to the sub-triangles formed by each cevian with the adjacent sides, leveraging a generalization of the angle bisector theorem. This approach establishes the equivalence between the classical side-ratio form and the trigonometric form without relying on areas or coordinates. Specifically, for cevian AD intersecting BC at D, the law of sines in triangles ABD and ACD yields the ratio

\frac{BD}{DC} = \frac{AB \cdot \sin \angle BAD}{AC \cdot \sin \angle CAD},

since the angles at D are supplementary and thus have equal sines.^[20] Analogous applications in triangles BCE and BEA for cevian BE intersecting CA at E, and in triangles CAF and CBF for cevian CF intersecting AB at F, produce

\frac{CE}{EA} = \frac{BC \cdot \sin \angle CBE}{AB \cdot \sin \angle ABE}, \quad \frac{AF}{FB} = \frac{AC \cdot \sin \angle ACF}{BC \cdot \sin \angle BCF}.

Multiplying these three relations gives the side-ratio form of Ceva's theorem,

\frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB} = 1,

where the side lengths cancel as

\frac{AB}{AC} \cdot \frac{BC}{AB} \cdot \frac{AC}{BC} = 1,

leaving the trigonometric condition

\frac{\sin \angle BAD}{\sin \angle CAD} \cdot \frac{\sin \angle CBE}{\sin \angle ABE} \cdot \frac{\sin \angle ACF}{\sin \angle BCF} = 1.

This equivalence holds because the side-ratio form implies concurrency (as proved by other methods, such as areas), and thus so does the trigonometric form.^[20] Assuming concurrency at an interior point O, the law of sines can also be applied directly in the three triangles AOB, BOC, and COA formed around O. In △AOB,

\frac{\sin \angle OAB}{\sin \angle OBA} = \frac{OB}{OA};

in △BOC,

\frac{\sin \angle OBC}{\sin \angle OCB} = \frac{OC}{OB};

and in △COA,

\frac{\sin \angle OCA}{\sin \angle OAC} = \frac{OA}{OC}.

The product of these ratios simplifies to 1, as the segment lengths OA, OB, and OC cancel:

\frac{\sin \angle OAB}{\sin \angle OBA} \cdot \frac{\sin \angle OBC}{\sin \angle OCB} \cdot \frac{\sin \angle OCA}{\sin \angle OAC} = 1.

This identity confirms the necessity of the trigonometric condition under concurrency and chains through the angular measures in the six smaller triangles adjacent to O (such as those involving the side-division points D, E, F). To connect back to the side ratios, the sines of angles at O relate via the full set of sub-triangles, but the earlier derivation provides the direct link.^[21]

Applications

Triangle Centers and Configurations

Ceva's theorem plays a fundamental role in verifying the concurrency of cevians associated with classical triangle centers, providing a unified criterion through the product of segment ratios equaling unity. For the centroid, the medians serve as cevians connecting each vertex to the midpoint of the opposite side, dividing each side in the ratio 1:1; thus, the product of these ratios is $1 \times 1 \times 1 = 1, confirming concurrency at the centroid, which divides each median in the ratio 2:1.^[22] Similarly, the angle bisectors concur at the incenter, where the division ratios on the sides are proportional to the adjacent side lengths: for the bisector from vertex A meeting side BC at D, \frac{BD}{DC} = \frac{AB}{AC} = \frac{c}{b}; from B to CA at E, \frac{CE}{EA} = \frac{a}{c}; and from C to AB at F, \frac{AF}{FB} = \frac{b}{a}; the product simplifies to \frac{c}{b} \cdot \frac{a}{c} \cdot \frac{b}{a} = 1.^[22] For the orthocenter, the altitudes as cevians require the trigonometric form of Ceva's theorem to verify concurrency at the orthocenter in acute triangles (with appropriate signed ratios for obtuse cases).^[22] This application highlights Ceva's versatility, extending beyond simple segment ratios to angular measures for perpendicular cevians. In more advanced configurations, Ceva's theorem confirms concurrency for notable points like the Brocard points, defined as the point where cevians from each vertex form equal angles \omega (the Brocard angle) with the respective sides; the trigonometric form of Ceva's theorem establishes their existence and uniqueness. Likewise, in Morley's trisector theorem, Ceva's theorem aids in proving concurrencies within the configuration of angle trisectors, where intersections form an equilateral triangle; the theorem verifies that the product of sine ratios for the trisecting cevians satisfies the concurrency condition in the subdivided angles.^[23] The converse of Ceva's theorem enables the construction of a concurrence point O given desired division ratios on the sides, provided their product equals 1: select points D, E, F on sides BC, CA, AB respectively such that \frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB} = 1, then the cevians AD, BE, CF intersect at O; this method allows systematic location of O by solving for one ratio when the others are specified, often using barycentric coordinates for precision.^[22] Isogonal conjugates exemplify Ceva's role in symmetric configurations, where for points P and Q, the cevians to P and to Q are reflections of each other over the angle bisectors; the Ceva ratios for Q are the reciprocals of those for P in the trigonometric form, ensuring \prod \frac{\sin \angle BAP}{\sin \angle CAP} = \prod \frac{\sin \angle BAQ}{\sin \angle CAQ}^{-1} = 1 for both, thus confirming dual concurrencies related by bisector reflections.^[22]

Relation to Menelaus' Theorem

Menelaus' theorem provides a condition for the collinearity of points on the sides of a triangle formed by a transversal line. Specifically, for a triangle ABC with a transversal line intersecting sides AB at F, BC at D, and CA at E, the signed ratios satisfy the relation \frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = -1.^[24] Ceva's theorem and Menelaus' theorem are complementary, with Ceva addressing the internal concurrency of three cevians (lines from vertices to opposite sides) as a configuration analogous to three transversals meeting at a point, in contrast to Menelaus' single transversal establishing collinearity.^[25] Their proofs are often dual in the projective plane, where incidence relations between points and lines are preserved under duality, transforming concurrence into collinearity and vice versa.^[26] A specific projective link arises in the complete quadrilateral, where applying Menelaus' theorem to one inscribed triangle implies Ceva's condition on another triangle within the same configuration through perspectivity, highlighting their interdependence in projective transformations.^[24] These theorems find joint application in proving Desargues' theorem on the perspectivity of triangles and in pole-polar relations, where duality maps poles to polars while maintaining the theorems' conditions.^[26] Both theorems derive from area-based or barycentric coordinate methods, where Ceva's product equals +1 for concurrent cevians and Menelaus' equals -1 for a transversal, the sign difference arising from orientation in the plane.^[25]

Generalizations

Trigonometric Form

The trigonometric form of Ceva's theorem provides a condition for the concurrency of cevians AD, BE, and CF in \triangle ABC using ratios of sines of angles at the vertices. Specifically, the cevians are concurrent if and only if

\left( \frac{\sin \angle BAD}{\sin \angle CAD} \right) \left( \frac{\sin \angle CBE}{\sin \angle ABE} \right) \left( \frac{\sin \angle ACF}{\sin \angle BCF} \right) = 1.

^[27] This form is derived by applying the law of sines in the adjacent triangles formed by each cevian.^[28] The trigonometric form offers advantages over the classical length-based version when direct measurement of side segments is challenging, but angular information is available. It is particularly effective in problems involving angle divisions like trisectors, as seen in proofs of Morley's theorem where the trisectors intersect to form an equilateral triangle, verified via the sine ratios. Additionally, it extends naturally to non-Euclidean settings, such as spherical triangles, where an analogous condition \sin BD \cdot \sin CE \cdot \sin AF = \sin DC \cdot \sin EA \cdot \sin FB holds for concurrency of great circle arcs.^[29]^[30] This trigonometric variant is equivalent to the classical Ceva's theorem in Euclidean geometry through the law of sines applied to the base segments of the adjacent triangles, which relates the sine ratios directly to length ratios. However, it stands independently in non-Euclidean geometries like spherical trigonometry, where length ratios do not directly translate due to curvature, but the sine condition persists.^[28]^[30] A simple verification occurs with angle bisectors, where each cevian divides the vertex angle equally, so \angle BAD = \angle CAD, \angle CBE = \angle ABE, and \angle ACF = \angle BCF. Thus, the sine ratios are all 1, and their product equals 1, confirming concurrency at the incenter.^[27]

Extensions to Simplexes and Polygons

Ceva's theorem extends naturally to higher-dimensional simplices, where the role of line segments (cevians) in a triangle is generalized to rays from vertices to points on opposite (n-1)-dimensional faces, concurrent at an interior point. In an n-simplex S with vertices v_0, \dots, v_n, consider points p_F on each facet F_i opposite v_i. These points induce a multipede if the cevians v_i p_{F_i} concur at a point p_S in the interior of S, satisfying a product condition over volume ratios of subregions (lobes) on the facets. Specifically, for every uniform or mixed m-fan of k-lobes on a facet, the product \prod_{z \in \mathbb{Z}/m\mathbb{Z}} \frac{\mathrm{Vol}(L_z)}{\mathrm{Vol}(M_z)} = 1, where L_z and M_z are the volumes of the respective lobes adjacent to p_F.^[31] For the tetrahedron, the 3-simplex, this manifests as four points on the triangular faces, with cevians from vertices to these face points concurrent if the product of signed area ratios on each face equals 1, generalizing the edge length ratios of the triangular case. This condition ensures the cevians meet at a single interior point, preserving the concurrency criterion through (n-1)-dimensional volumes rather than 1-dimensional lengths. A comprehensive extension confirming this for arbitrary n, using barycentric coordinates to define the multipede induced by any interior point, was established in 2021.^[31] Barycentric coordinates provide a unified framework for these extensions, as they generalize seamlessly from triangles to n-simplices, where the product condition on affine ratios along cevians remains invariant under affine transformations. This preserves the core of Ceva's theorem: the concurrency point corresponds to barycentric coordinates whose components satisfy the volume ratio products equaling 1 across all facets.^[31] Beyond simplices, Ceva's theorem generalizes to polygons, particularly convex polygons with an odd number of sides $2n+1, where cevians from each vertex to a point on the "opposite" side (defined cyclically) concur if a multivariate product of directed segment ratios equals 1. For a polygon P = [A_1 A_2 \dots A_{2n+1}], with points B_k on the side opposite A_k, the cevians A_k B_k are concurrent precisely when \prod_{k=1}^{2n+1} \frac{A_{n+k} B_k}{B_k A_{n+k+1}} = 1 (indices modulo $2n+1). This extends the triangular product to higher odd-sided figures, such as pentagons, using directed distances along sides.^[32]

References

[1]
Ceva's Theorem -- from Wolfram MathWorld
Given a triangle with polygon vertices A, B, and C and points along the sides D, E, and F, a necessary and sufficient condition for the cevians AD, BE, ...
[2]
[PDF] The Theorems of Ceva and Menelaus | OSU Math
Jun 19, 2014 · Ceva's theorem and Menelaus's Theorem have proofs by barycentric coordinates, which is effectively a form of projective geometry; see [Sil01], ...
[3]
[PDF] Exploring Geometry
This theorem has become known as Ceva's Theorem, in honor of Gio- vanni ... Proof: The first equality is a re-statement of the preceding theorem. The.<|control11|><|separator|>
[4]
[PDF] Ceva's Theorem
Grab point D and move it along segment BC . this lab, can you state the trigonometric form of Ceva's Theorem? * If you are not familiar with Cabri's tools, ...
[5]
[PDF] Contents
Aug 3, 1999 · In certain cases, Ceva's Theorem is more easily applied in the following trigonometric form. sinZCAP sinZAP B sinZABQ sinZQBC sinZBCR sinZRCA = ...
[6]
Cevian -- from Wolfram MathWorld
The condition for three general Cevians from the three vertices of a triangle to concur is known as Ceva's theorem. Cevians. Picking a Cevian point P in the ...
[7]
Ceva's theorem - PlanetMath
Mar 22, 2013 · Ceva's theorem. Let ABC A ⁢ B ⁢ C be a given triangle and P P any point of the plane. If X X is the intersection ...<|control11|><|separator|>
[8]
https://planetmath.org/cevastheorem
[9]
[PDF] Let's Teach More Accurate and Inclusive History! The Case of ...
Jul 2, 2025 · Ceva (1647–1734), it was actually first stated and proven in The Book of Completion by Ibn Hud (d. 1085), a medieval Islamic mathematician ...
[10]
[PDF] Chapter 4 - Concurrency of Lines in a Triangle
Definition 4.1 A cevian is a line segment which joins a vertex of a triangle with a point on the opposite side (or its extension). Using Ceva's Theorem we can ...
[11]
[PDF] Mass Point Geometry - Berkeley Math Circle
Sep 8, 2015 · Given a triangle, a cevian is a line segment from a vertex to a point on the interior of the opposite side. (The 'c' is pronounced as 'ch').
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[PDF] Euclid's Partition Problem and Ceva's Theorem
Definition 3. Segments are concurrent if they intersect at a common point. Theorem 1 (Ceva). The cevians AD, BE, CF, of △ABC ...
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Ceva's Theorem
Sep 10, 2011 · Three cevians of a triangle are concurrent if and only if Ceva's equation holds. Or in other words, if ( A ' B C ) , ( A B ' C ) , and ( A B ...
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[PDF] College Geometry
A line going through exactly one vertex of a triangle is called a. Cevian of the triangle. ... The pedal triangle of acute △ABC is △DEF. Perpendiculars AU ...<|control11|><|separator|>
[15]
Existence of the Orthocenter
The foot of an altitude also has interesting properties. Altitudes as Cevians. This is Corollary 3 of Ceva's theorem. Orthocenter as Circumcenter. The ...
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[PDF] Barycentric Coordinates in Olympiad Geometry - Evan Chen
Jul 13, 2012 · Ceva's Theorem guarantees that these will cancel cyclically, so we just compute the proper determinants, knowing that they will cancel ...
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[PDF] Barycentric Coordinates - Zachary Abel
Aug 17, 2007 · ... proof of Ceva's theorem, which states that cevians AD, BE, and. CF are concurrent if and only if the three ratios BD. DC. = ra, CE. EA. = rb, ...
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proof of trigonometric version of Ceva's theorem - PlanetMath
Mar 22, 2013 · The generalization of bisectors theorem states that Multiplying the three equations, and cancelling all segments on the right side gives the desired ...
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Trigonometric Form of Ceva's Theorem
The theorem states that, in \Delta ABC, three Cevians AD, BE, and CF are concurrent iff the following identity holds.
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Ceva's Theorem
Ceva's theorem is the reason lines in a triangle joining a vertex with a point on the opposite side are known as Cevians.
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[PDF] Triangles and Groups via Cevians - Western CEDAR
For a given triangle T and a real number ρ we define Ceva's triangle Cρ(T) to be the triangle formed by three cevians each joining a vertex of T to the point ...Missing: history | Show results with:history
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Of Looking and Seeing
Of Looking and Seeing: Ceva's theorem helps prove concurrencies in Morley's configuration. Plenty of near misses (or is it near hits).
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[PDF] Duality in projective geometry. The theorems of Menelaus, Ceva and ...
In Menelaus' theorem, using directed ratios changed the product from a ”1” to ”−1”. Is the same thing true for Ceva's theorem?
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Menelaus and Ceva - MathPages
For the same set of velocities, the Theorem of Ceva gives six lines that intersect in four points, one internal and three external, as shown below. APOLLO2.<|control11|><|separator|>
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[PDF] A Unified Proof of Ceva and Menelaus' Theorems Using Projective ...
Ceva and Menelaus theorems are well known. However, these theorems characterize a projec- tive property (concurrence in Ceva's theorem and collinearity in ...
[26]
[PDF] Geometry Unbound - Kiran S. Kedlaya
Mar 10, 2011 · In certain cases, Ceva's Theorem is more easily applied in the following form (“trig. Ceva”). Fact 5.1.2 (Ceva's theorem, trigonometric form).
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[PDF] Trigonometry and Analytic Tools in Olympiad Geometry Problems ...
Dec 11, 2023 · From the Law of Sines at the triangles PBD,PDC we find that sin∠BPD ... theorem, as well as Trigonometric Ceva. Exercise 7: Prove the ...
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[PDF] a direct trigonometric proof of morley's theorem
Sep 20, 2019 · We establish a direct short proof of Morley's theorem using trigonometry. 1. Introduction. Morley's theorem is one of the most beautiful ...
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[PDF] spherical trigonometry - Internet Archive
This theorem is analogous to Ceva's theorem * for a plane triangle. o. Conversely, when the points D, E, F in the sides of a spherical triangle are such that ...
[30]
[PDF] An Extension of Ceva's Theorem to n-Simplices
Ceva's theorem is a central result of post-Euclidean geom- etry. The theorem was first formulated and proved in the 11th century by Yusuf al-. Mu'taman ibn Hud ...Missing: Sesiano | Show results with:Sesiano
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[PDF] Extensions of some Ceva type theorems in polygons
The most known concurrence theorems in the triangle concern the concurrence of some important lines: the concurrence of the medians, the concurrence of the.Missing: tangential | Show results with:tangential