5 The hidden shape
Let’s have one final look at the Pappus configuration, but this time let’s pretend that we don’t know the conclusion.
The idea is to fix the triangle \(A\alpha X\) and look for Ceva configurations on it. As Ceva configurations are so simple, finding them isn’t very hard. We just have to choose another point that doesn’t lie on \(X\beta , A\beta\) or \(AX\), we call these the forbidden lines. For example, we can choose \(Z\), which doesn’t lie on the forbidden lines by the non-degeneracy data (i.e. the assumption that the points \(A,B,C,X,Y\) and \(Z\) are all distinct and \(AC \neq XZ\)). We represent this Ceva configuration by drawing a triangle labelled \(Z\).
Note: Even though we label the triangle by writing \(Z\) in its interior, that doesn’t imply that the point \(Z\) is in the interior of the triangle \(A\alpha X\). This figure is simply a visual way or recording the fact that \(Z\) gives a Ceva configuration on \(A\alpha X\).
From now on, for any given point \(P\), the \(P\)-configuration is the Ceva configuration obtained by considering the triangle \(A\alpha X\) together with the Ceva point \(P\). So, for example, the figure above depicts the \(Z\)-configuration. When introducing a new \(P\)-configuration we won’t provide an argument that \(P\) doesn’t lie on one of the forbidden lines. However, in each case an analogous argument to the one provided for \(Z\) could be made using the non-degeneracy data. Let’s turn our focus to the \(Y\)-configuration. The crucial realisation is that, as \(X,Y\) and \(Z\) are collinear, the \(Y\)-configuration defines the same intersection point along \(A\alpha\) as the \(Z\)-configuration. Recall that we colour Ceva points green and intersection points orange.
This is exactly the compatibility condition that allows us to glue these two Ceva configurations along \(A\alpha\). To symbolise this, we draw two triangles correspondingly glued together.
Similarly, the \(B\)-configuration can also be glued into the picture.
Adding the \(C\)-configuration is a bit trickier. Not only does it define the same intersection point along \(AX\) as the \(Y\)-configuration (since \(Y,\alpha\) and \(C\) are collinear), it also defines the same intersection point along \(\alpha X\) as the \(B\)-configuration (since \(A,B\) and \(C\) are collinear).
We can therefore glue the \(C\)-configuration to the \(Y\)-configuration along \(AX\) and also to the \(B\)-configuration along \(\alpha X\). To symbolise this we not only glue a triangle labelled \(C\) to the triangle labelled \(Y\), but also identify its \(\alpha X\) edge with the corresponding edge of the triangle labelled \(B\).
We represent the identification of different edges by colouring those edges with a common colour and indicating an orientation (which makes the identification unambiguous). In the same manner we add in the \(\gamma\)-configuration.
Here comes the final coup de grâce. Let’s consider the following triangulated surface.
Our above discussion demonstrates how we can equip each triangle on \(S\) (except the top one) with a Ceva configuration coming from the points \(C,Y,Z,B\) and \(\gamma\). Furthermore, our proposition on compatible Ceva configurations from the previous section tells us that there exists a point \(\tilde{\beta}\) such that the \(\tilde{\beta}\)-configuration is compatible with all its adjacent configurations.
Let \(P\), \(Q\) and \(R\) be the intersection points that lie on \(A\alpha , X\alpha\) and \(XA\) respectively. By definition, we have \(\tilde{\beta} = XP \cap AQ\). However, we also know that the \(\beta\)-configuration is compatible with the \(C\)-configuration (since \(X,\beta\) and \(C\) are collinear) and the \(Z\)-configuration (since \(A,\beta\) and \(Z\) are collinear). Therefore \(\beta = XP \cap AQ = \tilde{\beta}\).9 But that means the \(\beta\)-configuration is also compatible with the \(\gamma\)-configuration, which is equivalent to \(\alpha\), \(\beta\) and \(\gamma\) being collinear (since it implies that the lines \(\alpha\beta\) and \(\alpha\gamma\) share the points \(\alpha\) and \(R\) and therefore coincide). We’ve done it, we’ve proved Pappus’ theorem!
You might be wondering why, during our proof, we only saw symbolic depictions of \(A\alpha X\) being glued to itself. Why didn’t we just explicitly draw all the compatible Ceva configurations? The reason is that, if you start with a standard looking Pappus configuration (like this one) then all the glued-together Ceva configurations tend to look very chaotic. However, if you’re happy to start with a less standard looking Pappus configuration, you can explicitly draw all the resulting Ceva configurations into a coherent picture.
This figure is actually coded backwards: all the Ceva configurations are declared first and then the Pappus configuration is worked out. For this reason all the left-hand-side points are immobile, but moving the point labelled \(Z\) on the right-hand-side varies the initial Ceva configurations.
Now that the proof is complete, let’s take a moment to bask in just how beautiful it is. It almost feels like, once you’ve established Ceva’s theorem and written down the triangulated surface \(\mathcal{S}\), the theorem just proves itself. We didn’t really have to do anything, just interpret the consequences. Talking of \(\mathcal{S}\), let’s try and get a better idea of what it actually looks like, starting by labelling the glued edges and removing the triangulation.
We consider two paths on \(\mathcal{S}\): one, called \(d\), is obtained by following \(a\) and then \(c\), and the other, called \(e\), by following \(b\) and then the reverse of \(a\). Writing \(\mathcal{S}\) with respect to these paths gives the following.
If you don’t recognise \(\mathcal{S}\), check out this video. The thick blue lines in that video correspond to our paths, \(d\) and \(e\). That’s right, \(\mathcal{S}\) is a torus.
Personally, I find it cool that \(\mathcal{S}\) is both a relatively simple but topologically non-trivial surface. It’s deeply embedded into the theorem, yet so hard to find. Have another look at the statement of Pappus’ theorem. It’s hard to believe that all along there was a secret torus behind the scenes! Identifying \(\mathcal{S}\) also raises a natural question: what happens it we use a different surface? Surely things can’t get even better?
This argument is simply explaining why if two Ceva configurations are both compatible with two other Ceva configurations, then they coincide.↩︎