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Rhombus problem, with tangents and parallel (Posted on 2013-01-13) Difficulty: 3 of 5
ABCD is a rhombus. Take points E, F, G, H on sides AB, BC, CD, DA respectively so that EF and GH are tangent to the incircle of ABCD. Show that EH and FG are parallel.

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 3.0000 (1 votes)

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Solution Solution Comment 1 of 1
 
It is easy to show that EH and FG are parallel if 
and only if triangles HAE and FCG are similar.
 
Let O be the intersection of diagonals AC and BD
(also, it's the center of the incircle). Let E',
F', G', and H' be the points of tangency of the
incircle with the sides AB, BC, CD, and DA
respectively. Let I and J be the points of 
tangency of the incircle with EF and GH 
respectively. In the following a string of three
capital letter denotes an angle unless specified
otherwise. Let the twelve angles about O be
 
    u = H'OH = HOJ
    v = JOG  = GOG'
    z = G'OC = COF'
    x = F'OF = FOI
    y = IOE  = EOE'
    z = E'OA = AOH'.
 
Clearly, the following holds
 
    u + v + z = x + y + z = 90.              (1)
 
With algebra and a trig. identity (1) implies
 
    tan(x)+tan(z)     tan(v)+tan(z)
   --------------- = ---------------.        (2)
    tan(z)+tan(u)     tan(z)+tan(y)  
 
Multiplying both sides of (2) with r/r (where
r is radius of the incircle) gives
 
    r*tan(x)+r*tan(z)     r*tan(v)+r*tan(z)
   ------------------- = -------------------   
    r*tan(z)+r*tan(u)     r*tan(z)+r*tan(y)  
 
                     or
 
    |EE'| + |E'A|     |GG'| + |G'C|
   --------------- = ---------------   
    |AH'| + |H'H|     |CF'| + |F'F|  
 
               or
 
    |EA|     |GC|
   ------ = ------.   
    |AH|     |CF|  
 
This and HAE = FCG implies the triangles
are similar.
 
QED


  Posted by Bractals on 2013-01-14 13:07:32
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