Isogonal Line Lemma (updated)
by utkarshgupta, Jan 5, 2016, 7:14 AM
So I happened to stumble upon this lemma while solving a problem.
I know I should have used a diagram to prevent configuration thing but I don't really know how to use geogebra.
if someone can provide the diagram.
Lemma :
Let
and 
be two pairs of isogonal lines with respect to 
.
Let
and 
.
Then
are isogonal line with respect to
Now let us solve some problems using this lemma.
Problem 1 (India Postals 2015 Set 2) :
Let
be a convex quadrilateral. In the triangle 
let 
and 
be the incenter and the excenter opposite the vertex 
, respectively. In the triangle 
let 
and 
be the incenter and the excenter opposite the vertex , respectively. Show that the lines 
and 
, and the bisector of the angle 
are concurrent.
Problem 2 (USAMO 2008} :
Let
be an acute, scalene triangle, and let 
, 
, and 
be the midpoints of 
, 
, and 
, respectively. Let the perpendicular bisectors of and 
intersect ray 
in points 
and 
respectively, and let lines 
and 
intersect in point 
, inside of triangle . Prove that points 
, , , and all lie on one circle.
Problem
(China TST 2002) (Generalization of INMO 2003) :
Let
and 
be the intersections of opposite sides of a convex quadrilateral . The two diagonals meet at 
. Let 
be the foot of the perpendicular from to 
. Show that 
.
Problem
(Iran TST 2015)
is the altitude of triangle and 
is the reflection of 
trough the midpoint of 
. If the tangent lines to the circumcircle of at 
and 
, intersect each other at 
and the perpendicular line to 
at , intersects 
and 
at 
and 
respectively, prove that 
.
Problem
(Tournament of Towns Spring Senior A 2006)(given by Anant Mudgal)
In triangle, let 
be the bisector, and let be any point on and let 
meet 
again at 
. Let 
meet 
at and 
meet 
at 
. Prove that 
.
I know I should have used a diagram to prevent configuration thing but I don't really know how to use geogebra.
if someone can provide the diagram.
Lemma :
Let
and 
be two pairs of isogonal lines with respect to 
.Let
and 
.Then
are isogonal line with respect to 
intersect 
in 
respectively.
(configuration matters here)
our lemma.Now let us solve some problems using this lemma.
Problem 1 (India Postals 2015 Set 2) :
Let
be a convex quadrilateral. In the triangle 
let 
and 
be the incenter and the excenter opposite the vertex 
, respectively. In the triangle 
let 
and 
be the incenter and the excenter opposite the vertex , respectively. Show that the lines 
and 
, and the bisector of the angle 
are concurrent.Problem 2 (USAMO 2008} :
Let
be an acute, scalene triangle, and let 
, 
, and 
be the midpoints of 
, 
, and 
, respectively. Let the perpendicular bisectors of and 
intersect ray 
in points 
and 
respectively, and let lines 
and 
intersect in point 
, inside of triangle . Prove that points 
, , , and all lie on one circle.
is the symmedian of 
that is the isogonal conjugate of the median.
and perpendicular to 
and perpendicular to 
is the midpoint of 
.
are obviously isogonal conjugates w.r.t 
,
are isogonal conjugates.
is the symmedian.Problem
(China TST 2002) (Generalization of INMO 2003) :Let
and 
be the intersections of opposite sides of a convex quadrilateral . The two diagonals meet at 
. Let 
be the foot of the perpendicular from to 
. Show that 
.
is a harmonic pencil, so is 
that is 
is the angle bisector of 
and 
are isogonal lines with respect to 
and 
,
are isogonal lines with respect to 
Problem
(Iran TST 2015) 
is the altitude of triangle and 
is the reflection of 
trough the midpoint of 
. If the tangent lines to the circumcircle of at 
and 
, intersect each other at 
and the perpendicular line to 
at , intersects 
and 
at 
and 
respectively, prove that 
.
are concyclic,
and 
are isogonal lines with respect to 
,
)
and 
are also a pair of isogonal lines with respect to 
Problem
(Tournament of Towns Spring Senior A 2006)(given by Anant Mudgal) In triangle
, let 
be the bisector, and let be any point on and let 
meet 
again at 
. Let 
meet 
at and 
meet 
at 
. Prove that 
.This post has been edited 3 times. Last edited by utkarshgupta, Mar 4, 2016, 12:38 PM
,
be points inside so that 
are isogonal. Show that if 
and 
then 
are isogonal in 
and 
Then 
and 
which gives 
Similarly 
hence 
and 
swap under the Involution of reflection over the angle-bisector of 
(we use DDIT on the concave quadrangle) w.r.t. 
swap under this Involution. This means that 
and 
September 2017
February 2017