70. Colinearity with pole/polar - H\in PQ .
by Virgil Nicula, Jul 30, 2010, 2:56 AM
P1. Let 
be a triangle with orthocenter 
. The tangent lines from 
to the circle 
with diameter ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
touch it on 
and 
. Prove that 
.
Proof 1 (power of point - Sunken rock). Denote
- orthic triangle of 
, where 
, 
, 
and power 
of 
w.r.t. circle 
. Observe
that , 
, belong to the circle 
with diameter ![$[AM]$](//latex.artofproblemsolving.com/1/f/9/1f9b22599237fb6240a50b5f75e8f6ced1292374.png)
and 
is radical axis of , . From 
obtain that .
Remark. From the pairs of circles
, 
denote analogously the pairs of tangent points 
, 
of the tangents from 
, 
to 
, 
respectively. Thus 
, 
, 
and 
. Hence 
, i.e. the ponts , , 
, 
, 
, 
belong to same circle. is radical center for 
and also for 
. If denote 
, then 
and 
, where 
is midpoint of , what is another well-known remarkable property.
Proof 2 (pole/polar). Let the circumcircle
of , 
, 
for which 
and 
. Thus, 
, i.e. 
what is a characterization of 
- harmonical division, i.e. is harmonical conjugate of w.r.t. 
. Hence belongs to polar of w.r.t. , i.e. .
Proof 3 (metric). Suppose that
separates , . Denote 
so that 
and so that 
. Observe that 
and 
. Thus 
, i.e. 
, , 
are concurrently .
Lemma. Let
be a cyclic hexagon. Prove that 
.
Proof 4 (inversion). Let be midpoint of 
. Observe that , , belong to the circle with diameter 
. Apply an inversion with center at and radius 
. Because 
, we obtain that is the image of under this inversion, and , remains invariant. For the other hand, 
is transformed into the line containing , . Because 
we can conclude that .
P2. Let
be the incenter of the non-isosceles triangle 
and let 
be the tangency points of the incircle with the sides 
respectively.
The lines
and 
intersect in , the lines 
and 
in and the lines 
and intersect in 
. Prove that the lines 
and 
are perpendicular.
Proof Denote
. From the wellknown relations 
results 
,
i.e. the line is the image of the circumcircle with the diameter ![$[IP]$](//latex.artofproblemsolving.com/e/a/9/ea973f41f63c92568ec369e5487bc59ac5800094.png)
through the inversion with the pole I and the constant 
. Therefore, 
be a triangle with orthocenter 
. The tangent lines from 
to the circle 
with diameter ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
touch it on 
and 
. Prove that 
.Proof 1 (power of point - Sunken rock). Denote
- orthic triangle of 
, where 
, 
, 
and power 
of 
w.r.t. circle 
. Observethat
, 
, belong to the circle 
with diameter ![$[AM]$](http://latex.artofproblemsolving.com/1/f/9/1f9b22599237fb6240a50b5f75e8f6ced1292374.png)
and 
is radical axis of , . From 
obtain that .Remark. From the pairs of circles
, 
denote analogously the pairs of tangent points 
, 
of the tangents from 
, 
to 
, 
respectively. Thus 
, 
, 
and 
. Hence 
, i.e. the ponts , , 
, 
, 
, 
belong to same circle. is radical center for 
and also for 
. If denote 
, then 
and 
, where 
is midpoint of , what is another well-known remarkable property.Proof 2 (pole/polar). Let the circumcircle
of , 
, 
for which 
and 
. Thus, 
, i.e. 
what is a characterization of 
- harmonical division, i.e. is harmonical conjugate of w.r.t. 
. Hence belongs to polar of w.r.t. , i.e. .Proof 3 (metric). Suppose that
separates , . Denote 
so that 
and so that 
. Observe that 
and 
. Thus 
, i.e. 
, , 
are concurrently .Lemma. Let
be a cyclic hexagon. Prove that 
.Proof 4 (inversion). Let
be midpoint of 
. Observe that , , belong to the circle with diameter 
. Apply an inversion with center at and radius 
. Because 
, we obtain that is the image of under this inversion, and , remains invariant. For the other hand, 
is transformed into the line containing , . Because 
we can conclude that .P2. Let
be the incenter of the non-isosceles triangle 
and let 
be the tangency points of the incircle with the sides 
respectively.The lines
and 
intersect in , the lines 
and 
in and the lines 
and intersect in 
. Prove that the lines 
and 
are perpendicular.Proof Denote
. From the wellknown relations 
results 
,i.e. the line
is the image of the circumcircle with the diameter ![$[IP]$](http://latex.artofproblemsolving.com/e/a/9/ea973f41f63c92568ec369e5487bc59ac5800094.png)
through the inversion with the pole I and the constant 
. Therefore, 
This post has been edited 26 times. Last edited by Virgil Nicula, May 16, 2017, 8:38 AM
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