15580Re: A very simple geometry question..
- Oct 1, 2007--- In Hyacinthos@yahoogroups.com, "malisundaresan"
> Prove that the line joining the -circumcenter of a triangle to a
> vertex- is perpendicular to the line -joining the foot of
> adjoining sides.-Let E, F be the feet of the altitudes through
> Some sort of a HINT please....
B, C upon AC, AB respectively,
Let X be the midpoint of AH (H orthocenter)
X is the center of the circle through A, E, H, F.
XE = XF
The nine-point circle (N) passes through E and F
NE = NF (each = R/2)
Follows NX is the perpendicular bisector of EF.
In the triangle AOH :
X, N being the midpoints of AH, OH we have
NX is parallel to OA.
Together with the fact that NX is perpendicular to EF,
it now follows that OA is perpendicular to EF
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