Three Isosceles Triangles: What Is It About? A Mathematical Droodle

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Explanation 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!

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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 A, B, the orientation of the triangle and the angle α. It could be expressed as a complex linear combination of A and B:

 (1) X = (1 - c)A + cB,

where c = 1/2 + k·i, i2 = -1, and k being a real number. The real part of c is bound to equal 1/2 in order for the triangle ABX to be isosceles. By the construction,

 (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 + (c2 + (1-c)2)B + c(1 - c)C.

Now,

 c(1 - c) = (1/2 + k·i)(1/2 - k·i) = 1/4 + k2,

while

 c2 + (1-c)2 = (1/2 + k·i)2 + (1/2 - k·i)2 = 1/2 - 2k2.

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

1. Honsberger, In Pólya's Footsteps, MAA, 1999 