Slick Projectives!
by anantmudgal09, Sep 4, 2016, 6:15 PM
Dedicated to Utkarsh, a one-line proof of the isogonality lemma.
Lemma: Let
be five points in the general position in the plane. Suppose that 
are isogonal lines in angle 
and 
and 
. Then, 
are isogonal in angle .
Proof (communicated to me by user liberator):- By applying the dual of Desrague's involution theorem on the quadruple
we conclude that an involution swaps 
, 
and 
. Since are isogonal, the involution is in fact the reflection in the bisector of angle and the result follows.
I will try looking for more applications of this idea. There is a somewhat longer but projectively insightful proof using dynamic, but let's save it for another blog post.
Lemma: Let
be five points in the general position in the plane. Suppose that 
are isogonal lines in angle 
and 
and 
. Then, 
are isogonal in angle .Proof (communicated to me by user liberator):- By applying the dual of Desrague's involution theorem on the quadruple
we conclude that an involution swaps 
, 
and 
. Since are isogonal, the involution is in fact the reflection in the bisector of angle and the result follows.I will try looking for more applications of this idea. There is a somewhat longer but projectively insightful proof using dynamic, but let's save it for another blog post.
Probably this: In triangle 
lines 
from its midpoint. Then so do the lines 
.
September 2017
February 2017