Three Isosceles Triangles: What Is It About?
A Mathematical Droodle
What if applet does not run? 
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Copyright © 19962018 Alexander Bogomolny
Three Isosceles Triangles
The applet suggests the following theorem [Honsberger, p. 271]:
Suppose A, B, C are arbitrary points on a straight line, and α is a given angle. Assume also that similarly oriented isosceles triangles ABX and BCY have the same apex angles α at X and Y. If XYZ is an isosceles triangle with the apex angle at Z equal to α, but having orientation opposite to that of triangles ABX and BCY, then Z is always collinear with A, B, and C!
What if applet does not run? 
One simple solution makes use of the properties of complex numbers and their geometric interpretation. Thus I shall interpret all vertices and points of intersection as complex numbers in the plane of the diagram with an arbitrarily chosen origin.
Point X  the apex of an isosceles triangle ABX  is uniquely defined by the pair
(1)  X = (1  c)A + cB, 
where c = 1/2 + k·i,
(2)  Y = (1  c)B + cC, 
and also
(3)  Z = (1  c)Y + cX, 
Substituting (1)(2) into (3) gives:
(4)  Z = c(1  c)A + (c^{2} + (1c)^{2})B + c(1  c)C. 
Now,
c(1  c) = (1/2 + k·i)(1/2  k·i) = 1/4 + k^{2}, 
while

So, in fact, the linear combination (4) is real, i.e. having all real coefficients. Moreover, the three coefficients in (4) add up to 1. This exactly means that Z (the value of the combination) is collinear with A, B, and C.
Remark
It must be noted that the fact that (4) is real is independent of the requirement that the real part Re(c) of c equals 1/2. Which implies that the theorem remains valid if isosceles triangles are replaced with similar ones. Such a generalization has been suggested by Nathan Bowler who also supplied a very simple synthetic proof of the latter (and hence of the current theorem.) Nathan also came up with further generalization wherewith the third triangle is formed for any point X' on BX and any point Y' on BY. The apex Z' of such triangles is still collinear with A, B, and C.
References
 Honsberger, In Pólya's Footsteps, MAA, 1999
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny
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