Ceva's Theorem

A Proof


Let's begin by stating the theorem as Giovanni Ceva did back in 1678:

We start with triangle ABC, with three cevians drawn concurently through a given point, in this case, point P.

The proof of this theorem depends on the relationship between the area of a triangle, and its base and altitude. We are going to describe triangles which have the needed segments as bases in terms of their areas and altitudes.

Let's begin with the segments AY and YC, and their ratio. Let's look at some triangles that have these segments as bases, namely triangles ABY, and CBY.


One thing you need to notice is that both triangles have the same height (BH1). So when we express their areas in terms of base and height, we see the following:

When we write these values as the ratio, we see that the altitudes drop out leaving us with =.

But, the above triangles are not the only triangles with bases AY and YC. We can also use triangles APY and CPY.


We can use the same strategy as above to write . Now, since we now know that,

and that areas are additive, we can state that , or , using geometry's area axiom.

It is using the above steps and applying them to the other ratios of Ceva's theorem, that will allow us to rerwite the other two ratios as:

So, now, if we replace each of the ratios in Ceva's original equation, with the ones in term of areas of the triangles, we see that:

and that

so, we can conclude that

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