Question 1208063
<pre>

There is a fallacy in Ikleyn's proof above.  I will draw just enough of it to
show the fallacy.  While she is correct that triangles AOP and QOR are
congruent, QR is not necessarily parallel to AP, and therefore not necessarily
parallel to AB as she has stated. I do not see that she has proved it parallel.

We must rule out the possibility of the figure on the right below, and show that
the figure on the left is the correct one. She is correct that QR = AP = 3, but
she must show that QR is parallel to AP.  I think she can show this by making
use of the fact that QR = 3 and the converse of this well-known theorem:

<b><i>If a line parallel to one side of a triangle intersects the other two sides, 
then it divides the two sides proportionally. (The converse is also true.)</i></b>

If Ikleyn will correct this error in her proof, I will delete this post. So that
the student will learn of the corrected proof, I will make another post in
one of my other screen names, to inform him or her of the correction.  

{{{drawing(200,1200/7,-1,6,-1,5,
locate(2.9,1.9,O),
line(1,0,1,3),line(1,3,5,1),line(5,1,5,4),line(1,0,5,4),
locate(5,1,P),locate(5,4.5,A),locate(1,3.5,R), locate(.9,0,Q),

locate(3.3,2.4,alpha),locate(2.3,2.2,alpha),


locate(5.1,2.7,3),locate(.625,1.7,3)

)}}}{{{drawing(200,1200/7,-1,6,-1,5,

line(1.41886116991581033400,0.418861169915810334001,5,4),line(0.4701778718652965344,3.26491106406735173280,5,1),line(5,1,5,4),line(1.41886116991581033400,0.418861169915810334001,0.4701778718652965344,3.26491106406735173280),
locate(2.9,1.9,O),locate(5.1,2.7,3), locate(.625,1.9,3),

locate(3.3,2.4,alpha),locate(2.3,2.2,alpha),

locate(5,1,P),locate(5,4.5,A), locate(.4,3.8,R), locate(1.3,.44,Q)





)}}}

Below I have drawn the figure for the whole problem, but I purposely drew QR to
look not quite parallel to AB, which Ikleyn will prove it to be. She is welcome
to use my figure below.  

{{{drawing(300,300,0,12,0,10,
line(1.4,2.6,11,9),line(11,1,11,9),line(4.85,4.9,11,6),line(5.15,1.975,4.85,4.9),line(1.4,2.6,11,1), locate(1.4,2.6,C), locate(5,2,Q),locate(12,0,B),
locate(11.15,6.2,P),locate(11.15,9.2,A),locate(4.5,5.3,R), locate(11,1,B),
line(11,9,5.15,1.975), locate(8.05,5.45,O),locate(8.55,6.16,alpha),
locate(7.05,5.3,alpha),locate(11.15,3.6,5),locate(11.15,7.7,3),
locate(4.55,3.55,3)


 )}}}

Edwin</pre>