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
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Copyright © 19962018 Alexander Bogomolny