7 Incenter-related things every Geometry student should know
A line through the incenter and circumcenter
This line intersects the circumcircle at D and E, and the incircle at F and G.
r2 = DF x EG
A line through the incenter and midpoint of a side.
A line though the midpoint D and the incenter I intersects the altitude from C at F. The distance from F to C is equal to the radius of the incircle
Incenter cuts angle bisector into ratio
The incenter I cuts the angle bisector CD into two parts where
DI : IC = AB : AC + CB
Distance from Orthocenter to incenter and excenters
Let points E, F, and G be the excenters of triangle ABC and I the incenter. The sum of the squares of the distances from these centers to the orthocenter is triple the square of the diameter of the circumcircle.
OI2 + OD2 + OE2 + OF2 = 3 x (Circumcircle Diameter)2
Product of Distances from Incenter to Triangle Vertices
The product of the distances from the incenter to the triangle vertices is equal to the product of the square of the incircle diameter and the circumcircle radius
IA x IB x IC = d2 x R
The Distances from Incenter to the CIrcumcenter
This is Euler's Triangle Formula
The distance from the incenter to the circumcenter is given by
where R is the circumradius and r is the inradius.
Area of triangle based on inradius
The area of a triangle can be found by multiplying the inradius by the semiperimeter of the triangle.