n≥=3 lines lie in the same plane and are concurrent at point O.
If y = mx is the equation of a line passing through the origin O
(where m is the slope), then m_k labels the line y = tan(k*180°/n)*x
(for k = 0 to n-1). Note: If n is even and k = n/2, then line m_k is
perpendicular to line m₀. Point P (distinct from O) is an arbitrary point
in the plane of the n lines. F_i is the foot of the perpendicular from
point P to line m_i (for i = 0 to n-1).

Prove that F₀F₁...F_n-1 is a regular n-gon.
  

For i = 0 to n-1:

Since /PF_iO = 90⁰, all points F_i lie on a circle with diameter OP.

Since /F_iOF_i+1 = 180⁰/n, all chords F_iF_i+1 subtend that same angle

at O, on the circumference, so the chords are all equal.

Thus F₀F₁……F_n-1 is a regular n-gon

Posted by Harry on 2017-07-09 12:02:32