If you've studied geometry to a certain degree, of course you know about the four classic centers of triangles. But there are two more famous centers, not often taught in schools, called the Nagel and Gergonne points. They are related in that they both pertain to cevians, so here they are, in the same post.

First, what is an excircle? If you take a triangle and extend two of the sides, the circle tangent to those lines and the side of the triangle excircle or escribed circle.

Now, when you draw the cevians from each vertex of a triangle to the point on the other side where it's tangent to the excircle, you find that these cevians intersect at the Nagel point.

Similarly, the Gergonne point is when you draw the cevians from each vertex of the triangle to the point on the other side where it's tangent to the inscribed circle.



(And I totally didn't just violate a copyright there.)



To prove that the Gergonne point exists is just a simple application of Ceva's Theorem, which you can read about in my blog post here. (See that handy dandy link there? I didn't have to scour through pages of my blog to find that! I just clicked the category "Geometry". I love being organized. Now time to get disorganized!)

The Nagel point, however, is more interesting. To show that it exists, we need to know some more about excircles, as all we have now is a simple definition.

As you hopefully know, an inscribed circle is tangent to all three sides of a triangle, since its center is equidistant from the sides. That's because it lies on all three angle bisectors, which, as you recall, consist of all the points equidistant from the sides. Now, consider that a side of a triangle isn't just a segment, but rather a whole big line with the segment inside it. Therefore, the center of the inscribed circle isn't the only point that's equidistant from the sides.

Let's look at the angle bisector of an exterior angle. It has all the points equidistant from one side of a triangle and the extension of another. If we draw the bisector of a nearby exterior angle (like if you use AC's extension the first, use AB's), we then have an intersection point that's equidistant from all three sides, just like two interior angle bisectors. Such an outside concurrent point, which also, of course, lies on the third angle bisector, is then the center of one of three excircles.

Notice that each excircle is tangent to the sides, just as an incircle is. Thus, the Nagel point is the intersection of three cevians connecting each vertex of the triangle to the point of tangency of the corresponding excircle at the opposite side.

In the diagram below, we can see the point of tangency, X, of the excircle opposite point A.

[img]http://polymathematics.typepad.com/.a/6a00d8341bfda053ef010535f45b0e970c-800wi[/img]

See how the blue segments are congruent, just like the purple and green. Now, notice how if we go from A to C to X, that's the same length as A to C to P. Similarly, that is the same as A to B to Q, and A to B to X. Therefore, we can say that AB+BX = AC + CX. That means X isn't just the point of tangency of the excircle opposite A, but also the semiperimeter () point from A!

So now we can label our triangle like this:

[img]http://polymathematics.typepad.com/.a/6a00d8341bfda053ef010535edf257970b-pi[/img]

And from here, it's an easy peasy application of Ceva's.