The 3 distances of a point from each of the sides of the triangle.
If point P is (signed) distance Ta from side BC, Tb from side AC and Tc from side AB then its exact
trilinear coordinates are written as Ta:Tb:Tc. A point on the same side of BC as vertex A has positive distance, 0 if on the line and a negative distance if on the other side of that line; similarly for the other sides. So in the diagram here, Ta and Tb are positive and Tc is negative.

A point P is represented by the signed areas of the three subtriangles made by joining P to
vertices A, B and C, called the exact barycentric coordinates and
written as {area of PBC, area of PAC, area of PBA} often written as {α, β, γ} .

Its radius is one half of the circumcircle through all 3 vertices.
Feuerbach's Theorem of 1822 states that the ninepoint circle is tangent to the incircle and the three excircles.
Many other ETC centers lie on the Euler line. A list is given on Eric W Weisstein's "Euler Line" Mathworld page
 see Links and References at the foot of this page.
Highest . . . . . . . . . . Lowest  
unary + unary −  ^  * / % //  binary + binary − 
ETC Centers and Links
Use 6 9 13 to search for ETC centers sides: 



Select Coordinates to convert:  
to ETC Center X() 

© 2019 Dr Ron Knott  Back to Dr Knott's Maths HOME page 