207. Some interesting problems of Toshio Seimiya, Japan.
by Virgil Nicula, Jan 11, 2011, 10:06 AM
PP1 (T.S.). Let 
be a convex quadrilateral. Denote 
and the midpoints 
, 
, 
of ![$[BD]$](//latex.artofproblemsolving.com/3/2/6/3261e689901bce018ecdef47b9bc60a78ead3746.png)
, ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
,
![$[AC]$](//latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
respectively. Prove that ![$AL\cap IM\cap DN\ne\emptyset\iff AD\parallel BC\ \vee\ [ABCD]=2\cdot [BIC]$](//latex.artofproblemsolving.com/8/5/c/85c56db50903935f653bcd8aa22fdfc02512084f.png)
.
PP2 (T.S.). Let be a convex quadrilateral. Denote , the midpoints , of ![$[AB]$](//latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
, respectively
and the point
for which 
. Prove that 
is a trapezoid.
PP3 (T.S). Let
be an acute triangle. Construct outside of it the equilateral triangles 
, 
. Denote the intersections
, 
and 
. Prove that ![$[AFPG]=[BPC]\ \iff\ A=60^{\circ}$](//latex.artofproblemsolving.com/c/c/f/ccfece1622b8f636b7d0bc0650a17dae9e8463e2.png)
.
PP4 (own). In the convex quadrilateral denote 
. Prove that 
and 
have same centroid iff 
.
PP5 (own). In convex denote 
and the midpoints , , 
, 
of , ,
![$[CD]$](//latex.artofproblemsolving.com/e/7/0/e70960e9e5738a46ad23f794e796ef3cb4ad7e2c.png)
, ![$[DA]$](//latex.artofproblemsolving.com/7/b/e/7be1ee9448ef63ac6f1f6b8dd6982c30bad9bb31.png)
respectively. Denote 
. Prove that 
.
PP6. Let
with incircle 
Prove that 
so that 
is tangent to 
exists the relation 
be a convex quadrilateral. Denote 
and the midpoints 
, 
, 
of ![$[BD]$](http://latex.artofproblemsolving.com/3/2/6/3261e689901bce018ecdef47b9bc60a78ead3746.png)
, ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
,![$[AC]$](http://latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
respectively. Prove that ![$AL\cap IM\cap DN\ne\emptyset\iff AD\parallel BC\ \vee\ [ABCD]=2\cdot [BIC]$](http://latex.artofproblemsolving.com/8/5/c/85c56db50903935f653bcd8aa22fdfc02512084f.png)
.
of the vectorial system and denote 
for any 
. Therefore, 
. The lines 
, 
, 
are concurrently 
exist three real numbers 
, 
and 
so that
. From elimination of the parameters 
. Appear two cases :
![$[CID]=[AIB] \implies$](http://latex.artofproblemsolving.com/2/5/4/25498ae942c1bd2070f35e577f91932753caa429.png)
![$ [ACD]=[ABD]$](http://latex.artofproblemsolving.com/9/8/9/9891261e734c993d7c18c431756696fcf98f6ff0.png)
.
![$\frac {[ABCD]}{ [AIB]}=2\cdot \frac {[BIC]}{[AIB]}\implies$](http://latex.artofproblemsolving.com/8/1/9/819ff992a1bb50816ad323d8f68fefc3e1b77d58.png)
![$[ABCD]=2\cdot [BIC]$](http://latex.artofproblemsolving.com/5/6/3/5632fa7041c19b24569b4b895d90ce0e16589b6c.png)
.PP2 (T.S.). Let
be a convex quadrilateral. Denote , the midpoints , of ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
, respectivelyand the point
for which 
. Prove that 
is a trapezoid.
. The lines 
, 
, 
are concurrently 
. From elimination of the parameters 
![$\iff [CID]=[AIB]\iff [ABD]=[ACD]\iff$](http://latex.artofproblemsolving.com/c/f/6/cf6c2703a9ef982e606d46d27306d7f8b3e49365.png)
PP3 (T.S). Let
be an acute triangle. Construct outside of it the equilateral triangles 
, 
. Denote the intersections
, 
and 
. Prove that ![$[AFPG]=[BPC]\ \iff\ A=60^{\circ}$](http://latex.artofproblemsolving.com/c/c/f/ccfece1622b8f636b7d0bc0650a17dae9e8463e2.png)
.![$[AFPG]=[BPC]\iff $](http://latex.artofproblemsolving.com/0/c/b/0cbcf235b01ff0cc4e016391d9ad79a09d226a2d.png)
![$[ABG]=[BCF]\iff$](http://latex.artofproblemsolving.com/b/2/0/b2067a5099bbcd56745e15b9284ec4f4fec790c1.png)
.
and 
. Observe that 
because 
. From the relations 
obtain ![$[AFPG]=[BPC]\iff$](http://latex.artofproblemsolving.com/0/0/4/00490924c7e3810d9062ad7df3cacd8f50538716.png)
. Since 
obtain
. PP4 (own). In the convex quadrilateral
denote 
. Prove that 
and 
have same centroid iff 
.
.
exist the real numbers 
and 
so that 
. Hence 
and 
. From the elimination of 
, 
obtain that 
and 
and 
. Observe that 
.PP5 (own). In convex
denote 
and the midpoints , , 
, 
of , ,![$[CD]$](http://latex.artofproblemsolving.com/e/7/0/e70960e9e5738a46ad23f794e796ef3cb4ad7e2c.png)
, ![$[DA]$](http://latex.artofproblemsolving.com/7/b/e/7be1ee9448ef63ac6f1f6b8dd6982c30bad9bb31.png)
respectively. Denote 
. Prove that 
.
and exist the real numbers 
, 
so that 
.
and 
. Thus, exist 
so that 
,
, where 
and
. Since 
, 
)
and 
. PP6. Let
with incircle 
Prove that 
so that 
is tangent to 
exists the relation 
This post has been edited 59 times. Last edited by Virgil Nicula, Dec 10, 2017, 9:39 AM
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