Friendly Kiepert's Perspectors
What if applet does not run? 
Consider a configuration of two squares ACB_{c}B_{a} and BCA_{c}A_{b} with a common vertex C. Bottema's theorem asserts that the midpoint M of the segment A_{b}B_{a} is independent of C. The point M also serves as the apex of two rightangled (at M) isosceles triangles: AMB and A_{c}MB_{c}. In the proof of Bottema's theorem we have established an identity
(1)  (A_{b} + B_{a})/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:
(t_{l}) 

and
(t_{u}) 

Define M_{l} and M_{u} as the midpoints of X_{l}Y_{l} and X_{u}Y_{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
(t'_{l}) 

and
(t'_{u}) 

To prove that M_{l}, M_{u}, and C are collinear, we only have to show that the ratio
K = (m_{l}  g) / (m_{u}  g)
is real. However from (t'_{l})
(2_{l})  m_{l}  g = (a + b)/2 + i·t·(b  a)/2  g. 
On the other hand, from (t'_{u})
(2_{u})  m_{u}  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 X_{l}Y_{l}X_{u}Y_{u} and angles M_{l}BA and X_{l}CB are equal. This proof elucidates the meaning of t:
It follows that the Kiepert perspectors K(f) and
References
Bottema's Theorem
 Bottema's Theorem
 An Elementary Proof of Bottema's Theorem
 Bottema's Theorem  Proof Without Words
 On Bottema's Shoulders
 On Bottema's Shoulders II
 On Bottema's Shoulders with a Ladder
 Friendly Kiepert's Perspectors
 Bottema Shatters Japan's Seclusion
 Rotations in Disguise
 Four Hinged Squares
 Four Hinged Squares, Solution with Complex Numbers
 Pythagoras' from Bottema's
 A Degenerate Case of Bottema's Configuration
 Properties of Flank Triangles
 Analytic Proof of Bottema's Theorem
 Yet Another Generalization of Bottema's Theorem
 Bottema with a Product of Rotations
 Bottema with Similar Triangles
 Bottema in Three Rotations
 Bottema's Point Sibling
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Copyright © 19962017 Alexander Bogomolny
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