# Harmonic division (1)

Problem 1. Let $\triangle ABC$ be a triangle. The incircle of triangle $\triangle ABC$ touches side $BC$ at $A'.$ Let segment $AA'$ meet the incircle again at $P.$ Segments $BP,CP$ meet the incircle at $M,N,$ respectively. Show that lines $AA',BN,CM$ are concurrent.

Problem 2. Given acute triangle $ABC$ with $AB>AC$, let $M$ be the midpoint of $BC$. $P$ is a point in triangle $AMC$ such that $\angle MAB=\angle PAC$. Let $O,O_1,O_2$ be the circumcenters of $\triangle ABC,\triangle ABP,\triangle ACP$ respectively. Prove that line $AO$ passes through the midpoint of $O_1 O_2$.

Problem 3. Let $ABCD$ be a cyclic kite (i.e. $BD$ is a perpendicular chord onto the diameter $AC$) and $M$ the midpoint of $AD$. The perpendicular from $C$ onto $BM$ intersects $AD$ at $P$. Prove that $BP$ is tangent to the circle $\odot (ABC)$.

Problem 4. In triangle $ABC$, let $I$ be the incenter and let $I_a$ be the excenter opposite $A$. Suppose that $II_a$ meets $BC$ and the circumcircle of triangle $ABC$ at $A_0$ and $M$, respectively. Let $N$ be the midpoint of arc $MBA$ of the circumcircle of triangle $ABC$. Let lines $NI$ and $NI_a$ intersect the circumcircle of triangle $ABC$ again at $S$ and $T$, respectively. Prove that $S$, $T$, and $A_0$ are collinear.

Problem 5. Point $M$ lies on diagonal $BD$ of parallelogram $ABCD$. Line $AM$ intersects side $CD$ and line $BC$ at points $K$ and $N$, respectively. Let $C_1$ be the circle with center $M$ and radius $MA$ and $C_2$ be the circumcircle of triangle $KCN$. $C_1, C_2$ intersect at $P$ and $Q$. Prove that $MP$ and $MQ$ are tangent to $C_2$.

Problem 6. In an acute-angled triangle $ABC$, $AD,BE$ and $CF$ are the altitudes and $H$ the orthocentre. Lines $EF$ and $BC$ meet at $N$. The line passing through $D$ and parallel to $FE$ meets lines $AB$ and $AC$ at $K$ and $L$, respectively. Prove that the circumcircle of the triangle $NKL$ bisects the side $BC$.

Problem 7. Let $ABC$ an acute triangle so that $AB\not= AC$ and let $H$ the orthocenter of $ABC$. Points $D$ and $E$ are in the sides $AB,AC$ respectively so that $D,H,E$ are collinear and $AE=AD$. Prove that if $M$ is the midpoint of $BC$, then $MH$ is perpendicular to the straight line that joins $A$ with the second intersections (different of $A$) of the circumcircles of the triangles $AED$ and $ABC$.