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Math Forum » Discussions » Math Topics » geometry.pre-college.independent

Topic: kites
Replies: 5   Last Post: Nov 14, 1997 2:46 PM

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Mike de Villiers

Posts: 38
Registered: 12/6/04
Re: kites
Posted: Nov 14, 1997 2:46 PM
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Dear Eileen
Not sure whether you had a chance to investigate my "intuitive" guesses
regarding cyclic & circum quads (quad circumscribed around a circle) I
mentioned earlier. I just quickly checked it on Sketchpad and found the
following:

(1) If the point of intersection of the diagonals of a cyclic quad is
reflected over its sides, the resulting quad is a circum quad (ie. its
angle bisectors are concurrent at the intersection of the diagonals of
the cyclic quad)

Unfortunately the dual result which I expected to be true, namely:

(2) If the point of intersection of the diagonals of a circum quad is
reflected over its sides, the resulting quad is a cyclic quad

is NOT true, as the perpendicular bisectors of the sides of the resulting
quad are NOT concurrent, except in the special cases you've mentioned.
These perpendicular bisectors, however, pass through the vertices of the
original circum quad - an interesting result in itself.

I've not yet proved these (I'm under loads of final year marking, etc),
but do not think they'll be too difficult to prove. Please let me know in
the meantime if you or any of your students come up with proofs. Happy
exploring!

Regards
Michael de Villiers
Univ Durban-Westville
South Africa





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