[obm-l] A reference and proof

["Luis Lopes" <llopes@xxxxxxxxxxxxx>]

Sauda,c~oes,

O problema abaixo foi proposto numa
outra lista.

[]'s
Luís

Friends of Hycianthos:

Which is the reference and the proof of the problem:

Be ABC triangle rectangle in A, are BE and CD bisecting of
angles B and C. Let us considerer the segment ED and is M
the point of cut of height AH1 with ED. Proof that AM
measures the radius of the inscripte circle of ABC?
Thanks

Ricardo


>From: "Nikolaos Dergiades" <ndergiades@yahoo.gr>
>Reply-To: Hyacinthos@yahoogroups.com
>To: <Hyacinthos@yahoogroups.com>
>Subject: Re: [EMHL] A reference and proof
>Date: Sat, 13 Jul 2002 21:52:08 +0300
>
>Dear Ricardo,  I don't know a reference.
>
>A quick proof I can think is the following:
>If s is the semiperimeter, E the area of ABC
>F is the point of contact of AC and the incircle
>and r = inradius then
>the equation of the line DE in normals is
>x = y + z   because D = [1,1,0],  E = [1,0,1]
>If x, y, z are the actual normals of the point M
>(distances of M from the sides of ABC) then x = AH - AM ,
>y = AMcosC = AM*b/a,    z = AMcosB = AM*c/a
>and from x = y + z
>  we get   a*(AH - AM) = AM*b + AM*c
>
>or a*AH = AM*(a+b+c)  or  2*E = 2AM*s
>
>or 2*r*s = 2AM*s  or  r = AM.
>
>A synthetic proof must prove that the triangle AMF is
>isosceles or that MF is parallel to BE
>
>Best regards
>Nikos Dergiades


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