TikZ for geometric drawings?

Hi all,

I am starting to learn TikZ and I wonder whether I could use TikZ for 
geometric constructions like, say, the "nine points circle" of a 
triangle (EPS appended: mp-exemple.1).

Denis Roegel has shown how to do it in MetaPost in
http://www.gutenberg.eu.org/pub/GUTenberg/publicationsPDF/39-roegel.pdf

More precisely, here are things I cannot do (for now) in TikZ:
-- given a triangle ABC, how do I draw the perpendicular from A to BC?
I am looking for a kind of counterpart to MetaPost's syntax
G=whatever[B,C]=whatever[A,A+((C-B) rotated 90)];
-- how can I measure the distance between two constructed points 
(something like "r=arclength(I--N);" in MetaPost, to find out the radius 
of the circle)?

Thanks in advance for your hints and best wishes,
--

-- 
Daniel Flipo

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