Ceva's Theorem

Pronunciation: /sɛvʌz ˈθi ər əm/ Explain

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Manipulative 1 - Ceva's Theorem Created with GeoGebra.

Ceva's theorem states that; given a triangle ABC, and points D, E, and F that lie on lines AB, BC, and CA respectively, lines AE, BF and DC are concurrent if and only if (AD/DB)*(BE/EC)*(CF/FA)=1.[1]

The theorem is named after Giovanni Ceva (1647-1734), who proved it his 1678 work De lineis rectis. However it was proved much earlier by Yusuf Al-Mu'taman ibn Hud, an eleventh-century king of Zaragoza.

Proof of Ceva's Theorem

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Manipulative 2 - Proof of Ceva's Theorem Created with GeoGebra.
StepDiscussionJustification
0 Let ABC be a triangle. Let D be a point but not an endpoint on AB, E be a point but not an endpoint on BC, F be a point but not an endpoint on CA such that the segments AE, BF and CD are concurrent. Let P be the point where AE, BF and CD are concurrent. These are the criteria.
1 Draw a line through C parallel to AB. Label this line c.
2 Extend line segment AE. Label this line e. Label the intersection of lines c and e as A'.
3 Extend line segment BF. Label this line f. Label the intersection of lines c and f as B'.
4 Angle AEB and angle A'EC are congruent. Angles AEB and A'EC are opposite angles.
5 Angles A'CE and ABC are congruent Since BC is a transversal of parallel lines AB and c, angles A'CE and ABC are congruent.
6 Angles AA'C and A'AB are congruent. Since BC is a transversal of parallel lines AB and c, angles A'CE and ABC are congruent.
7 Triangles ABE and A'CE are similar triangles. Since AEB is congruent with A'EC, A'CE is congruent with ABC and AA'C is congruent with A'AB, triangles ABE and A'CE are similar triangles.
8 Triangles BAF and B'CF are similar triangles. Triangles BAF and B'CF are similar by an argument similar to steps 4-6.
9 The following equalities hold: BE/EC = AB/CA' and CF/FA = CB'/BA. The ratio of one corresponding side of a similar triangles to another corresponding side are equal.
10 Now multiply the respective sides of the equations in step 10 to get (BE/EC)*(CF/FA)=(AB/CA')*(CB'/BA)=(CB'/A'C). This uses the multiplicative property of equality and the substitution property of equality.
11 Triangle ADP is similar to triangle A'CP. Triangle ADP is similar to triangle A'CP by an argument similar to steps 4-6.
12 Triangle BDP is similar to triangle B'CP. Triangle BDP is similar to triangle B'CP by an argument similar to steps 4-6.
13 The following equalities hold: CP/DP=A'C/AD and CP/DP=CB'/DB The ratio of one corresponding side of a similar triangles to another corresponding side are equal.
14 This gives AD/DB=A'C/CB This uses the transitive property of equality and the multiplicative property of equality.
15 Multiplying the equation from step 10 with the equation from step 14 gives (AD/DB)*(BE/EC)*(CF/FA)=1'

Q.E.D.

This uses the multiplicative property of equality.

Converse

StepDiscussionJustification
1 Suppose that E,F,D are points on BC,CA and AB respectively satisfying (AD/DB)*(BE/EC)*(CF/FA)=1. These are the criteria.
2 Let Q be the intersection of AE with BF and D' be the intersection of CQ with AB.
3 Since AE, BF and CD' are concurrent, (AD'/D'B)*(BE/EC)*(CF/FB)=1 and (AD'/D'B)=(AD/DB). blank space
4 Step #3 implies D = D', so AE, BF, and CD are concurrent.

Q.E.D.

blank space
Table 1: Proof of Ceva's Theorem. Proof courtesy Yark. Licensed under Creative Commons Attribution 2.5 license.

References

  1. Godfrey, C. and Siddons, A.W.. Modern Geometry. pp 46-52. www.archive.org. Cambridge University Press. 1908. Last Accessed 8/6/2018. http://www.archive.org/stream/moderngeometry00godfrich#page/46/mode/1up/search/Ceva. Buy the book
  2. Coxeter, H.M.S. and Greitzer, S.L.. Geometry Revisited. 1st edition. pp 4-6. The Mathematical Association of America. 1967. Last Accessed 8/6/2018. Buy the book

More Information

  • McAdams, David E.. Cevian. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 6/27/2018. http://www.allmathwords.org/en/c/cevian.html.

Cite this article as:

McAdams, David E. Ceva's Theorem. 6/28/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/cevastheorem.html.

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Revision History

6/25/2018: Removed broken links, updated license, implemented new markup, updated GeoGebra apps. (McAdams, David E.)
1/9/2010: Added "References". (McAdams, David E.)
11/21/2008: Initial version. (McAdams, David E.)

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