Friendly Kiepert's Perspectors

Consider a configuration of two squares ACB_cB_a and BCA_cA_b with a common vertex C. Bottema's theorem asserts that the midpoint M of the segment A_bB_a is independent of C. The point M also serves as the apex of two right-angled (at M) isosceles triangles: AMB and A_cMB_c. In the proof of Bottema's theorem we have established an identity

a and b are complex number representations of vertices A and B, respectively.

We have the following generalization due to Floor van Lamoen.

Fix a real number t and construct four points:

and

Define M_l and M_u as the midpoints of X_lY_l and X_uY_u. Then the three points C, M_l and M_u are collinear.

The proof generalizes that of Bottema's theorem. Following the latter it is using complex numbers. a and b have been already introduced as complex numbers corresponding to points A and B. Let g correspond to C, and m_l, m_u, x_l, x_u, y_l, y_u correspond to M_l, M_u, X_l, X_u, Y_l, and Y_u.

(t_l) and (t_u) could be rewritten as

and

To prove that M_l, M_u, and C are collinear, we only have to show that the ratio

is real. However from (t'_l)

On the other hand, from (t'_u)

It is thus clear that K = t, a real number.

(The applet suggests a different proof based on the observations that X_lY_l||X_uY_u and angles M_lBA and X_lCB are equal. This proof elucidates the meaning of t: t = tan(X_lCB) = tan(M_lAB), whereas 1/t = tan(X_uCA_c) = tan(M_uA_cB_c).)

It follows that the Kiepert perspectors K(f) and K(p/2 - f) are related by friendship.

References

1. F. van Lamoen, Friendship Among Triangle Centers, Forum Geometricorum 1 (2001), pp. 1-6.

Copyright © 1996-2008 Alexander Bogomolny