## Incircles

In Euclidean Geometry, an incircle is the largest circle inside a triangle that is tangent to all three sides of the triangle. From the above discussion, though this exists for all triangles in Euclidean Geometry, the same cannot be said for Taxicab Geometry.

Because a taxicab circle is a square, it contains four vertices. This makes it hard for it to be tangent to all three sides of a triangle because there is one too many vertices. Therefore, a taxicab incircle is completely contained within a triangle with three of its vertices touching the sides of the triangle:

Because a taxicab circle is a square, it contains four vertices. This makes it hard for it to be tangent to all three sides of a triangle because there is one too many vertices. Therefore, a taxicab incircle is completely contained within a triangle with three of its vertices touching the sides of the triangle:

This definition clearly shows that there can be more than one incircle for certain types of triangles. However, there are cases where there is only one unique incircle. This occurs when the triangle is an inscribed triangle:

An interesting case, as seen in the first figure above, is when a triangle has one of its sides on a separator line, or one of the lines of slope 1 or -1. In this case, there will always be both an incircle and a circumcircle.

Finding incircles in Euclidean Geometry translates nicely to Taxicab Geometry. If a triangle has a taxicab incircle, the Euclidean incenter will be the taxicab incenter:

Finding incircles in Euclidean Geometry translates nicely to Taxicab Geometry. If a triangle has a taxicab incircle, the Euclidean incenter will be the taxicab incenter:

For more information about incircles, read the article, by Ermis, Gelisgen, and Kaya at the following link: http://master.grad.hr/hdgg/kog_stranica/kog16/03-12.pdf