High School Olympiads Forum 9
by franzliszt, Jul 23, 2024, 8:23 PM
High School Olympiads Forum 9 wrote:
Let 
be an acute triangle with circumcircle 
and intouch triangle 
. A circle is drawn tangent to 
at 
and to minor arc 
of at 
. Define 
and 
similarly. Prove that lines 
, 
, 
are concurrent.
be an acute triangle with circumcircle 
and intouch triangle 
. A circle is drawn tangent to 
at 
and to minor arc 
of at 
. Define 
and 
similarly. Prove that lines 
, 
, 
are concurrent.Proof. Let
be the midpoint of 
containing 
and define 
similarly. By Shooting Lemma, 
. In Barycentric Coordinates with reference triangle with 
we have 
and 
where we computed the coordinates of 
using the midpoint formula on normalized 
. We can compute similarly. We claim that the concurrency point is 
.Verify that
![\begin{align*}\begin{vmatrix}0&s-c&s-b\\a^2(s-a) & b^2(s-b) & c^2(s-c)\\ a^2&-b^2+bc&-c^2+bc\end{vmatrix}&=\frac{a^2}{4}\begin{vmatrix}0&a+b-c&a-b+c\\-a+b+c & b^2(a-b+c) & c^2(a+b-c)\\ 1&-b^2+bc&-c^2+bc\end{vmatrix}\\&=0+(a+b-c)[c^2(a+b-c)-(-a+b+c)(-c^2+bc)]+(a-b+c)[(-a+b+c)(-b^2+bc)-b^2(a-b+c)]\\&=0\end{align*}](http://latex.artofproblemsolving.com/d/f/5/df55b0afe316789bcdcfaaed94f9010f03b26dc2.png)
where we simplified the last part with a little help.We can use the exact same process to show that
lies on 
and 
completing the proof.
July 2024
June 2024