A Hard Geometry
by utkarshgupta, Apr 16, 2016, 5:54 AM
It took me so long to solve. Only if I knew a few projective theorems.
Problem : (RMM 2013 Problem 3)
Let
be a quadrilateral inscribed in a circle 
. The lines 
and 
meet at 
, the lines 
and 
meet at 
, and the diagonals 
and 
meet at 
. Let 
be the midpoint of the segment 
, and let 
be the common point of the segment 
and the circle . Prove that the circumcircle of the triangle 
and are tangent to one another.
Solution :
Let
be the center of 
.
Let
and 
.
Let
be the reflection of in 
Lemma :
Proof :
Well known
Obviously,
is a straight line (radical axis are concurrent).
Let
Then, since is the polar od , 
Thus and 
are straight lines..
We know that
and 
are cyclic quadrilaterals.
Hence
It is well known that
Since lies on the polar of ,
Also since is a straight line; is the midpoint of 
and 
Since is the reflection of about , is the reflection of about 
Hence
That is
Using the above results,
is a cyclic pentagon.
Now invert with radius
and centre .
Obviously
The cyclic pentagon implies,
and 
Thus one of
maps to .
Since
are tangent to 
;
and 
are tangent to 
.
QED.
Problem : (RMM 2013 Problem 3)
Let
be a quadrilateral inscribed in a circle 
. The lines 
and 
meet at 
, the lines 
and 
meet at 
, and the diagonals 
and 
meet at 
. Let 
be the midpoint of the segment 
, and let 
be the common point of the segment 
and the circle . Prove that the circumcircle of the triangle 
and are tangent to one another.Solution :
Let
be the center of 
.Let
and 
.Let
be the reflection of in 
Lemma :
Proof :
Well known
Obviously,
is a straight line (radical axis are concurrent).Let
Then, since
is the polar od , 
Thus
and 
are straight lines..We know that
and 
are cyclic quadrilaterals.
Hence
It is well known that
Since
lies on the polar of ,
Also since
is a straight line; is the midpoint of 
and 
Since
is the reflection of about , is the reflection of about 
Hence
That is
Using the above results,
is a cyclic pentagon.Now invert with radius
and centre .Obviously
The cyclic pentagon implies,
and 
Thus one of
maps to .Since
are tangent to 
;
and 
are tangent to 
.QED.
September 2017
February 2017