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Topic: feet of altitudes and angles
Replies: 5   Last Post: Dec 1, 1997 12:41 AM

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Floor van Lamoen

Posts: 32
Registered: 12/6/04
Re: feet of altitudes and angles
Posted: Nov 27, 1997 8:29 PM
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Annie Fetter wrote:
>
> My grandmother, who used to be a math teacher, read this problem long ago
> and has always wondered about a proof. (She was never 100% sure that it
> was really true until I modelled the problem in Sketchpad for her. Now she
> is a believer.) I haven't tried to prove it yet, but told her I would
> throw it out to the wolves and see what people came up with.
>
> Prove that the angles formed by connecting the feet of the altitudes of a
> triangle are themselves bisected by the altitudes.


Hi Annie,

Your grandmother was right when you take an acute triangle. When you have
an obtuse triangle you should change the angle-bisectors of two angles in
the outer angle-bisectors. Two sides do function as the 'normal'
bisectors then. Of course in a right triangle the triangle of the feet is
degenerate.

Back to the acute case, let A'B'C' be the feet of A, B and C:

C

B'
A'






A C' B


Since <BB'C = <BC'C = 90 there is a circle passing through BC'B'C'. In
this circle we can see that <CC'B'=<CBB'=90-C.
In a similar way we can see that <CC'A'=90-C.
Hence CC' bisects angle A'C'B'.
Of course the same argumentation holds for BB' and AA'.

Regards,
Floor van Lamoen





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