195. Semicircle. Nice application of harmonical division.
by Virgil Nicula, Dec 22, 2010, 8:49 AM
Proposed problem. Let ![$[CD]$](//latex.artofproblemsolving.com/e/7/0/e70960e9e5738a46ad23f794e796ef3cb4ad7e2c.png)
be a chord of the circle 
with the diameter 
so that doesn't separate 
and 
. Denote 
so that 
and 
such that 
. Prove that 
bisects angle 
.
Proof 1. I denote shortly
for an harmonical division.
quadrilaterals 
are cyclic 
An equivalent enunciation. Let
be a triangle for which the projection of 
to 
belongs to the side ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
. Denote the foot
of the-bisector, 
for which 
and 
. Prove that 
Proof 2. Denote the intersections
and 
. It is well-known that the division 
is harmonically, i.e. 
.
Apply the Menelaus' theorem to the transversal
and the triangle 
But 
. From 
But and for the projection 
of the point to the line exists
the same relation
, i.e. 
. In conclusion, [/hide]
Proof 3. Denote
and 
such that 
. Thus, 
are conxurrently 
In conclusion,
since the points
are concyclically obtain that 
Proof 4. Let
be on the circle 
so that 
are collinearly in that order. Let 
and 
intersect again
at
and 
respectively. Apply the Pascal's theorem to 
and find that 
. By the down lemma the polar of 
passes through 
Denote
and 
Observe that 
and 
So that 
is the orthocenter
of
Hence 
. Furthermore, the polar of also passes through 
, 
and . By the same lemma
, , and 
are collinearly. It follows that 
is an altitude of 
. By Blanchet's theorem 
Lemma. In cyclic quadrilateral
, let 
and 
. Then 
, and 
each lie on the polar of 
[/size]
![$[CD]$](http://latex.artofproblemsolving.com/e/7/0/e70960e9e5738a46ad23f794e796ef3cb4ad7e2c.png)
be a chord of the circle 
with the diameter 
so that doesn't separate 
and 
. Denote 
so that 
and 
such that 
. Prove that 
bisects angle 
.Proof 1. I denote shortly
for an harmonical division.
quadrilaterals 
are cyclic
An equivalent enunciation. Let
be a triangle for which the projection of 
to 
belongs to the side ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
. Denote the foot of the
-bisector, 
for which 
and 
. Prove that 
Proof 2. Denote the intersections
and 
. It is well-known that the division 
is harmonically, i.e. 
.Apply the Menelaus' theorem to the transversal
and the triangle 
But 
. From 
But and for the projection 
of the point to the line existsthe same relation
, i.e. 
. In conclusion, [/hide]Proof 3. Denote
and 
such that 
. Thus, 
are conxurrently 
In conclusion,since the points
are concyclically obtain that 
Proof 4. Let
be on the circle 
so that 
are collinearly in that order. Let 
and 
intersect again at
and 
respectively. Apply the Pascal's theorem to 
and find that 
. By the down lemma the polar of 
passes through 
Denote
and 
Observe that 
and 
So that 
is the orthocenterof
Hence 
. Furthermore, the polar of also passes through 
, 
and . By the same lemma, , and 
are collinearly. It follows that 
is an altitude of 
. By Blanchet's theorem 
Lemma. In cyclic quadrilateral
, let 
and 
. Then 
, and 
each lie on the polar of 
[/size]This post has been edited 30 times. Last edited by Virgil Nicula, Nov 22, 2015, 5:56 PM
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