Ceva's theorem

From Free net encyclopedia

Ceva's theorem is a very popular theorem in elementary geometry. Given a triangle ABC, and points D, E, and F that lie on lines BC, CA, and AB respectively, the theorem states that lines AD, BE and CF are concurrent if and only if

There is also an equivalent trigonometric form of Ceva's Theorem, that is, AD,BE,CF concur iff <math>\frac{\sin\angle BAD}{\sin\angle CAD}\times\frac{\sin\angle ACF}{\sin\angle BCF}\times\frac{\sin\angle CBE}{\sin\angle ABE}=1</math>.

It was first proven by Giovanni Ceva in his 1678 work De lineis rectis.

[edit]

Proof

Suppose <math>AD</math>, <math>BE</math> and <math>CF</math> intersect at a point <math>O</math>. Because <math>\triangle BOD</math> and <math>\triangle COD</math> have the same height, we have

<math>\frac{|\triangle BOD|}{|\triangle COD|}=\frac{BD}{DC}.</math>

Similarly,

<math>\frac{|\triangle BAD|}{|\triangle CAD|}=\frac{BD}{DC}.</math>

From this it follows that

<math>\frac{BD}{DC}= \frac{|\triangle BAD|-|\triangle BOD|}{|\triangle CAD|-|\triangle COD|} =\frac{|\triangle ABO|}{|\triangle CAO|}.</math>

Similarly,

<math>\frac{CE}{EA}=\frac{|\triangle BCO|}{|\triangle ABO|},</math> and <math>\frac{AF}{FB}=\frac{|\triangle CAO|}{|\triangle BCO|}.</math>

Multiplying these three equations gives

as required. Conversely, suppose that the points <math>D</math>, <math>E</math> and <math>F</math> satisfy the above equality. Let <math>AD</math> and <math>BE</math> intersect at <math>O</math>, and let <math>CO</math> intersect <math>AB</math> at <math>F'</math>. By the direction we have just proven,

Comparing with the above equality, we obtain

Adding 1 to both sides and using <math>AF'+F'B=AF+FB=AB</math>, we obtain

Thus <math>F'B=FB</math>, so that <math>F</math> and <math>F'</math> coincide (recalling that the distances are directed). Therefore <math>AD</math>, <math>BE</math> and <math>CF</math>=<math>CF'</math> intersect at <math>O</math>, and both implications are proven.

[edit]