ISL 2009 G4
by utkarshgupta, Apr 25, 2016, 11:51 AM
This proof is for Anant who asked me for a coordinate bash in the shouts 
I wasn't able to solve it synthetically (at least last year)
Problem : (ISL 2009)
Given a cyclic quadrilateral
, let the diagonals 
and 
meet at 
and the lines 
and 
meet at 
. The midpoints of 
and 
are 
and 
, respectively. Show that 
is tangent at to the circle through the points , and .
Proposed by David Monk, United Kingdom
Solution :
We will work in the Cartesian plane.
will mean that the 
,
coordinates of the point 
are 
respectively.
The slope of a line
will be denoted by 
.
Without loss of Generality,
,
,
Let
So set
Using
lie on the opposite sides of , and 
.
Now easy calculations yield
Observe that the given problem is equivalent to proving directed angles,
That is
Now this is just easy calculations (they are actually easy as most of the things cancel out).
I wasn't able to solve it synthetically (at least last year
)Problem : (ISL 2009)
Given a cyclic quadrilateral
, let the diagonals 
and 
meet at 
and the lines 
and 
meet at 
. The midpoints of 
and 
are 
and 
, respectively. Show that 
is tangent at to the circle through the points , and .Proposed by David Monk, United Kingdom
Solution :
We will work in the Cartesian plane.
will mean that the 
,
coordinates of the point 
are 
respectively.The slope of a line
will be denoted by 
.Without loss of Generality,
,
,
Let
So set
Using
lie on the opposite sides of , and 
.
Now easy calculations yield
Observe that the given problem is equivalent to proving directed angles,
That is
Now this is just easy calculations (they are actually easy as most of the things cancel out).
This post has been edited 3 times. Last edited by utkarshgupta, Apr 25, 2016, 11:54 AM
September 2017
February 2017