Side-Splitting Triangle Ratios
Side-Splitting Triangle Ratios
Going clockwise around a triangle, place a point on each side at position of the side's length. If segments are drawn from those points to the opposite vertex, the resulting inner triangle has area with ratio -4x(y-x)-x(y-x) when compared to the area of the original triangle. For example, when each side is divided by thirds, the inner triangle is exactly one-seventh the area of the outer triangle.
x
y
2
y
2
y
This is a special case of Routh's theorem, which was proven by Edward Routh in 1896.