227. IMO ShortList 2003 (problem 5).
by Virgil Nicula, Feb 22, 2011, 7:22 PM
Proposed problem. Let 
be an isosceles triangle with 
, whose incentre is 
. Let 
be a point on the circumcircle of the triangle 
lying inside the triangle. The lines through parallel to 
and 
meet 
at 
and 
, respectively. The line through parallel to
meets and at 
and 
, respectively. Prove that the lines 
and 
intersect on the circumcircle of the triangle 
Proof. The enunciation of this problem conceals vainly in its debut that the circumcircle of
is the circle with the diameter 
,
where
is the 
- exincenter of 
. Denote : the circumcircle 
of the triangle ; the circle 
with the diameter ![$[II_{c}]$](//latex.artofproblemsolving.com/4/d/3/4d360c2d7cf645c4c380fd8f59ddcee5827c26fd.png)
;
the circumcircles
, 
of the isosceles trapezoids 
, 
respectively ; the second intersection point 
between 
and the circle ; the intersection 
Prove easily that the lines , are the tangents from the point to the circle .
From the relation
obtain 
i.e. the parallelograms 
and 
are similarly as 
.
.
.
.
Variation on the same theme. Let be a fixed 
-isosceles triangle and let mobile 
, 
so that 
and exists 
for which
. Construct 
for which 
Ascertain the geometrical locus of 
.
be an isosceles triangle with 
, whose incentre is 
. Let 
be a point on the circumcircle of the triangle 
lying inside the triangle
. The lines through parallel to 
and 
meet 
at 
and 
, respectively. The line through parallel to meets
and at 
and 
, respectively. Prove that the lines 
and 
intersect on the circumcircle of the triangle 
Proof. The enunciation of this problem conceals vainly in its debut that the circumcircle of
is the circle with the diameter 
,where
is the 
- exincenter of 
. Denote : the circumcircle 
of the triangle ; the circle 
with the diameter ![$[II_{c}]$](http://latex.artofproblemsolving.com/4/d/3/4d360c2d7cf645c4c380fd8f59ddcee5827c26fd.png)
;the circumcircles
, 
of the isosceles trapezoids 
, 
respectively ; the second intersection point 
between 
and the circle
; the intersection 
Prove easily that the lines , are the tangents from the point to the circle .From the relation
obtain 
i.e. the parallelograms 
and 
are similarly as 
.
.
.
.Variation on the same theme. Let
be a fixed 
-isosceles triangle and let mobile 
, 
so that 
and exists 
for which
. Construct 
for which 
Ascertain the geometrical locus of 
.
where the point 
is the This post has been edited 11 times. Last edited by Virgil Nicula, Nov 22, 2015, 2:51 PM
August 2018
January 2018