Salt piles up in an interesting way when poured onto a triangle. Why does that happen?

As the salt piles up and reaches a maximum height, new grains of salt will fall off toward whichever side of the triangle is closest. We can show that points on an angle bisector are equidistant from the rays that form the angle. So salt grains that land on an angle bisector will balance and not fall toward either side. This is why we see ridges form in the salt.

As we might conjecture from the salt example, all three angle bisectors in a triangle meet at a single point, called the triangle’s incenter. To see why this is true, consider any two angle bisectors in a triangle. The point where they meet is the same distance from the first and second sides, and is also the same distance from the second and third sides. Therefore, the point is the same distance from all sides, so the third angle bisector must also go through this point.

In the images, segments and are angle bisectors. This means that, for angle , point is the same distance from ray as it is from ray . In triangle , point is the same distance from all three sides of the triangle—it’s the triangle’s incenter.

Angle QRS bisected by line T. Line from point T to line Q creates 90 degree angle. Line from point T to line S creates 90 degree angle. These segments are equivalent.