Ceva's theorem

by JISOOK OH

 

The line segment joining a vertex of a triangle to any given point on the opposite side is called a cevian. Thus if D, E, F are points on the respective sides BC, CA, AB of triangle ABC, the segments AD, BE, CF are cevians. This term comes from the name of the Italian mathematician Giovanni Ceva, who published in 1678 the following very useful theorem:

If three cevians AD, BE, CF, one through each vertex of a triangle ABC, are concurrent, then .

 

 

 

When we say that three lines (or segments) are concurrent, we mean that they all pass through one point, say P. To prove Ceva's theorem, we recall that the areas of triangles with equal altitudes are proportional to the bases of the triangles. Referring to above figure, we have

.
Smilarly,

Now, if we multiply these, we find



The converse of this theorem holds also:

If three cevians AD, BE, CF satisfy , then they are concurrent.

To see this, suppose that the first two cevians meet at P, as before, and that the third cevian through this point p is CF'. Then, by Ceva's theorem,

But we are assuming

Hence



F' coincides with F, and we have proved that AD, BE, CF are concurrent.

 

Click here for the GSP file


When P is inside triangle ABC, the ratio of the areas of triangle ABC and triangle DEF is always greater than or equal 4.

And when points D,E,F are midpoints of BC, AC, AB respectively, the ratio is equal to 4.

 

 

 

Click here for the GSP file


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