Isosceles triangle the hypotenuse to the right-angled vertex) divides the right triangle into two isosceles triangles. This is because the midpoint of the hypotenuse is the center of the circumcircle of the right triangle, and each of the two triangles created by the partition has two equal radii as two of its sides. Similarly, an acute triangle can be partitioned into three isosceles triangles by segments from its circumcenter, but this method does not work for obtuse triangles, because the circumcenter lies outside the triangle. Generalizing the partition of an acute triangle, any cyclic polygon that contains the center of its circumscribed
Incircle and excircles of a triangle The touchpoint opposite A is denoted "T", etc. This Gergonne triangle "TTT" is also known as the contact triangle or intouch triangle of "ABC". Its area is where formula_56, formula_57, formula_58 are the area, radius of the incircle and semiperimeter of the original triangle, and formula_7, formula_8, formula_9 are the side lengths of the original triangle. This is the same area as that of the extouch triangle. The three lines "AT", "BT" and "CT" intersect in a single point called the Gergonne point, denoted as "Ge" - "X(7)". The Gergonne point lies in the open orthocentroidal disk punctured at its