Friendly Kiepert's Perspectors

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

(1)(Ab + Ba)/2 = (a + b)/2 + (b - a)·i/2,

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:

(tl)
Xl = Xl(t) = B + t·(Ab - B),
Yl = Yl(t) = A + t·(Ba - A)

and

(tu)
Xu = Xu(t) = Ac + 1/t·(Ab - Ac),
Yu = Yu(t) = Bc + 1/t·(Ba - Bc).

Define Ml and Mu as the midpoints of XlYl and XuYu. Then the three points C, Ml and Mu 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 ml, mu, xl, xu, yl, yu correspond to Ml, Mu, Xl, Xu, Yl, and Yu.

(tl) and (tu) could be rewritten as

(t'l)
xl = b - i·t·(g - b),
yl = a + i·t·(g - a)

and

(t'u)
xu = (g + i·(g - a)) + 1/t·(g - a),
yu = (g - i·(g - b)) + 1/t·(g - b).

To prove that Ml, Mu, and C are collinear, we only have to show that the ratio

K = (ml - g) / (mu - g)

is real. However from (t'l)

(2l) ml - g = (a + b)/2 + i·t·(b - a)/2 - g.

On the other hand, from (t'u)

(2u) mu - g = g - i·(b - a)/2 - g/t + (a + b)/2t - g
= 1/t·((a + b)/2 + i·t·(b - a)/2 - g)

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

(The applet suggests a different proof based on the observations that XlYl||XuYu and angles MlBA and XlCB are equal. This proof elucidates the meaning of t: t = tan(∠XlCB) = tan(∠MlAB), whereas 1/t = tan(∠XuCAc) = tan(∠MuAcBc).)

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.

Bottema's Theorem

  1. Bottema's Theorem
  2. An Elementary Proof of Bottema's Theorem
  3. Bottema's Theorem - Proof Without Words
  4. On Bottema's Shoulders
  5. On Bottema's Shoulders II
  6. On Bottema's Shoulders with a Ladder
  7. Friendly Kiepert's Perspectors
  8. Bottema Shatters Japan's Seclusion
  9. Rotations in Disguise
  10. Four Hinged Squares
  11. Four Hinged Squares, Solution with Complex Numbers
  12. Pythagoras' from Bottema's
  13. A Degenerate Case of Bottema's Configuration
  14. Properties of Flank Triangles
  15. Analytic Proof of Bottema's Theorem
  16. Yet Another Generalization of Bottema's Theorem
  17. Bottema with a Product of Rotations
  18. Bottema with Similar Triangles
  19. Bottema in Three Rotations
  20. Bottema's Point Sibling

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