Mesmerising Isogonals
by MathPassionForever, Sep 27, 2019, 2:59 PM
Finally what you expect from me: a GEOMETRY blog post!!
So I was having a conversation with Naruto.D.Luffy and we came up with this as a result of a problem's generalisation.
![[asy]
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Problem: Perpendiculars
are dropped on line 
through 
and perpendiculars 
are dropped on lime 
, which is the isogonal conjugate of with respect to 
such that 
lie on , 
lie on . 
is the midpoint of 
and 
is the foot of -altitude on . Then prove that:
(i)
is cyclic with circumcenter .
(ii)
and 
are cyclic.
And this is the question from where we got this:
Edit:
Supercali and BOBTHEGR8 helped me study this configuration even more.
Let
be intersections of 
with 
as shown in the diagram. Then:
1)
and have similicenter .
2) Let
. Then 
are collinear, where the last point is the -Humpty point.
So I was having a conversation with Naruto.D.Luffy and we came up with this as a result of a problem's generalisation.
![[asy]
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Problem: Perpendiculars
are dropped on line 
through 
and perpendiculars 
are dropped on lime 
, which is the isogonal conjugate of with respect to 
such that 
lie on , 
lie on . 
is the midpoint of 
and 
is the foot of -altitude on . Then prove that:(i)
is cyclic with circumcenter .(ii)
and 
are cyclic.
and 
are similar and so are 
and 
.
.
on 
and we're done!
and 
pass though 
,
and now the problem is trivial by EGMO lemmas!And this is the question from where we got this:
Elnino2k wrote:
Given the circle 
and fixed, moves on . The circles with diameter 
and 
intersect at 
. Let 
be the intersection of 
with 
respectively where 
is midpoint of . Suppose that 
intersects 
at 
and 
intersects 
at .
Prove that
passes through a fixed point as moves.
and fixed, moves on . The circles with diameter 
and 
intersect at 
. Let 
be the intersection of 
with 
respectively where 
is midpoint of . Suppose that 
intersects 
at 
and 
intersects 
at .Prove that
passes through a fixed point as moves.Edit:
Supercali and BOBTHEGR8 helped me study this configuration even more.
Let
be intersections of 
with 
as shown in the diagram. Then:1)
and have similicenter .2) Let
. Then 
are collinear, where the last point is the -Humpty point.This post has been edited 25 times. Last edited by MathPassionForever, Apr 1, 2020, 6:43 PM
September 2019