Bottema with Similar Triangles

What Might This Be About?

Problem

Assume triangles $BET,$ $DAT,$ and $DFC$ are similar as shown below.

Bottema with similar triangles

Prove that $K,$ the midpoint of $EF$ is independent of the position of $D.$

Solution

Bottema with similar triangles

Assuming the point names as complex numbers, the similarity of the three triangles could be expressed as

$\displaystyle\frac{T-E}{B-E}=\frac{T-A}{D-A}=\frac{C-F}{D-F}=u,$

for a complex number $u.$ From here,

$\displaystyle\frac{(T+C)-(E+F)}{(B+D)-(E+F)}=u,$

implying

$\displaystyle\frac{(A+C)-(E+F)}{(A+B)-(E+F)}=u,$

which exactly means that $E+F$ and, with it, $\displaystyle\frac{E+F}{2}$ is independent of $D.$

Acknowledgment

This problem is a follow-up of the one posted by Dao Thanh Oai (Vietnam) at the CutTheKnotMath facebook page. The simple solution above has been posted by Leo Giugiuc (Romania) who also observed that Dao's problem extends to the one above. Here's Dao's formulation:

Triangles $BET,$ $TAD,$ $DFC$ are similar isosceles. Prove that the midpoint $K$ of $EF$ is independent of the position of $D.$

General Bottema's theorem with extra rotations - Dao's formulation with isosceles triangles

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