268. Saraghin contest, 2011.
by Virgil Nicula, Apr 23, 2011, 3:00 PM
Proposed problem 20. Let 
be a cyclical and tangential convex quadrilateral with the incircle 
and the
circumcircle
. Denote the midpoints 
, 
of its diagonals 
, 
. Prove that 
and 
.
Proof (metric). because ![$[AMB]+[CMD]=[ANB]+[CND]=[AIB]+[CID]=\frac S2=\frac{rs}{2}$](//latex.artofproblemsolving.com/7/7/8/7788e52aba2b5a86edc7eaa56284a5708e7cb6f8.png)
, where 
and the geometrical locus of the mobile point
for which ![$[ALB]+[CLD]=k$](//latex.artofproblemsolving.com/4/d/f/4df50307deff24ce3f0c70056faf11531796396b.png)
(constant) is a line. Denote 
, 
. Apply the theorem of median in :
. Therefore, 
, what is truly because 
.
Remark. Are well-known and the following relations :
and 
(the Durrande's relation).
The power of
w.r.t. 
is 
and 
. See and here
Proposed problem 21. Let
be an 
-right triangle with incenter 
and circumcircle 
. Denote: midpoints and of the sides ![$[AB]$](//latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
and ![$[AC]$](//latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
respectively;
intersections
and 
; second intersections 
, 
of the circle with 
, 
respectively. Prove that 
.
Proof (metric). Observe that![$\boxed{S=[ABC]=s(s-a)=(s-b)(s-c)=\frac 12\cdot bc}$](//latex.artofproblemsolving.com/5/c/d/5cd2e90a8216a52cb433354775a957f2b1d312d9.png)
and 
, where 
.
Denote
, where 
and 
. Prove easily that 
. For 
,
apply the Menelaus' theorem to the transversals 
.
In conclusion,
, what is truly.
be a cyclical and tangential convex quadrilateral with the incircle 
and thecircumcircle
. Denote the midpoints 
, 
of its diagonals 
, 
. Prove that 
and 
. Proof (metric).
because ![$[AMB]+[CMD]=[ANB]+[CND]=[AIB]+[CID]=\frac S2=\frac{rs}{2}$](http://latex.artofproblemsolving.com/7/7/8/7788e52aba2b5a86edc7eaa56284a5708e7cb6f8.png)
, where 
and the geometrical locus of the mobile point
for which ![$[ALB]+[CLD]=k$](http://latex.artofproblemsolving.com/4/d/f/4df50307deff24ce3f0c70056faf11531796396b.png)
(constant) is a line. Denote 
, 
. Apply the theorem of median in :
. Therefore, 
, what is truly because 
.Remark. Are well-known and the following relations :
and 
(the Durrande's relation).The power of
w.r.t. 
is 
and 
. See and hereProposed problem 21. Let
be an 
-right triangle with incenter 
and circumcircle 
. Denote: midpoints and of the sides ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
and ![$[AC]$](http://latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
respectively;intersections
and 
; second intersections 
, 
of the circle with 
, 
respectively. Prove that 
. Proof (metric). Observe that
![$\boxed{S=[ABC]=s(s-a)=(s-b)(s-c)=\frac 12\cdot bc}$](http://latex.artofproblemsolving.com/5/c/d/5cd2e90a8216a52cb433354775a957f2b1d312d9.png)
and 
, where 
.Denote
, where 
and 
. Prove easily that 
. For 
,
apply the Menelaus' theorem to the transversals 
.In conclusion,
, what is truly.This post has been edited 47 times. Last edited by Virgil Nicula, Nov 22, 2015, 8:35 AM
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