Problem
GEO-B3-M05-P014 Distances from the Lemoine Point
#14
★★★☆☆ Level 3 of 5
Let \(K\) be the Lemoine point of triangle \(ABC\), and let its distances to \(BC,CA,AB\) be \(x,y,z\). Prove that \(x:y:z=BC:CA:AB\).
Inspired by Prasolov special points geometry method
C. Hint 1. Use barycentric coordinates of the Lemoine point.
D. Hint 2. To pass to distances from sides, convert barycentrics to trilinears.
The Lemoine point has barycentric coordinates \((a^2:b^2:c^2)\), where \(a=BC\), \(b=CA\), \(c=AB\). Trilinear coordinates are proportional to distances from the sides, and the conversion is \((u:v:w)_{\text{bar}}=(a x:b y:c z)\).
Thus from \((a x:b y:c z)=(a^2:b^2:c^2)\), we get \(x:y:z=a:b:c\). Hence the distances from the Lemoine point to the sides are proportional to the corresponding side lengths.
A good entry point to trilinears without overload.