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I have been trying to make some videos of ruler and compasses constructions.

The symmedian point is fairly well known on this list. This is the best unbiased linear estimate assuming zero mean identically distributed uncorrelated errors for three lines of position.  It is also tha maximum of the probability if in addition teh errors are assumed Gaussian. So the next step is to construct something that helps understand the family of ellipses with constant sum of squared distance to the lines. In this video I constrict the major and minor axes of the ellipses, using the fact that the major axis goes through the incentre of similitude of the orthocentric and circumcentric circles, the triangle centre numbered X(1344) on Kimberling's Encyclopedia of Triangle Centres.

https://youtu.be/JS4ZgwZlAcc

Let me know what you think. I am still not good at making these videos. I have got a better pair of compasses and some coloured pencils and that will make it a bit clearer if I have another go.