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## Proof of the generalization of Brianchon theorem

Problem is proposed by Dao Thanh Oai.

Brianchon generalization

## 3 mixtilinear incircles have a common tangent

Problem (me): Given triangle $ABC$ with its circumcircle $(O)$. Let $L$ be an arbitrary point on arc $BC$ which does not contain $A$. Prove that the $A$-mixtilinear incircle of triangle $ABC$, $P$-mixtilinear incircles of triangles $PAB$ and $PAC$ have a common tangent.

Proof: See here

## Coaxal circles

Problem. Given a triangle $ABC$ with its circumcenter $O$. Let $d$ be an arbitrary line in the plane. $d$ intersects $BC, CA, AB$ at $X, Y, Z$, respectively. Let $H$ be the projection of $O$ on $d$. Prove that 3 circles $(AXH), (BYH), (CZH)$ are coaxal.

Solution.

Lemma: Given a triangle $ABC$ with its circumcircle $(O)$. Let $P$ be an arbitrary point in the plane. A line $d$ through $P$ and perpendicular to $OP$ intersects $(BPC), (CPA), (APB)$ again at $X, Y, Z$, respectively. Then $AX, BY, CZ$ are concurrent.

Proof:

We prove this lemma in case $d$ cuts $(O)$ and I will leave another case for the readers.
Let $J, K, L$ be the intersections of $AZ$ and $CX, BX$ and $AY, BC$ and $KJ$, respectively.
According to Desargues’s theorem, $AX, BY, CZ$ are concurrent iff $J, K, L$ are collinear.
We have $\angle ZAB=\angle ZPB=\angle BCJ$ then$J \in (O)$. Similarly, $K\in (O).$
Let $M, N$ be the intersections of $d$ and $(O)$. Note that $P$ is the midpoint of $MN$.
Since $LM.LN=LB.LC=LP.LX$ then $(NMLX)=-1$. We get $XJ.XC=XK.XB=XM.XN=XL.XP$ or $B,K,L,P$ and $P,L,C,J$ are concyclic.
Then $\angle PK=180^o-\angle PBX=180^o-\angle PCJ=180^o-\angle PLJ$. This means $K, L, J$ are collinear. We are done.
Back to our problem:

Let $A', B', C'$ be the second intersections of $(AHX), (BHY), (CHZ)$ and $(O)$, respectively. The idea is to show that $AA', BB', CC'$ concur at $H'$.
Let $M, N$ be the conserve point of $H$ and $X$, respectively.
The inversion with center $O$, power $R^2$ maps $(AHX)$ to $(AMN)$ then $A'\in (AMN)$.
Let $I$ be the intersection of $MN$ and $XH$. We have $\angle XNM=\angle XHM=90^o$ then $IM.IN=IX.IH$. This means $I$ lies on the radical axis of $(AHX)$ and $(AMN)$, or lies on $AA'$.
Let $K$ be the intersections of the tangents of $(O)$ at $B$ and $C$ then $M,N,K$ are collinear.
We get $N\in (OBC)$. So $XN.XO=XB.XC=XI.XH$ or $I$ lies on $(HBC)$. Applying the lemma above we obtain $AA', BB', CC'$ concur at $H'$. Therefore 3 circles $(AHX), (BHY), (CHZ)$ have the same radical axis $HH'$. We are done.

This is a nice problem I found a long time ago. It’s easy but nice. You can solve it by using inversion.

Problem:  Given a circumscribed quadrilateral $ABCD$. $AC$ intersects $(ABD)$ and $(CBD)$ again at $E, F$, respectively. Prove that $FBED$ is also a circumscribed quadrilateral.

## On the figure of IMO 2009 P2 Generalization

For the generalization, see the file below:

IMO P2 generalization

Problem 1: Given two triangles $ABC$  and $XYZ$ with their circumcircle $(O)$. $YZ$ intersects $AB, AC$ at $P, N$, respectively. Let $J$ be the projection of $O$ on $YZ$. Let $K, L$ be the midpoints of $BN, CP. A'$ be the reflection of $J$ wrt $KL$. Similarly we define $B', C'$. Prove that:

1 (Nguyen Van Thanh).  $A'$ lies on the 9-point circle of triangle $ABC$.

2.  Two triangles $XYZ$ and $A'B'C'$ are similar.

Problem 2: Given a triangle $ABC$ and its circumcircle $(O)$. Let $d$ be an arbitrary line.  $d$ intersects $BC, CA, AB$ at $X, Y, Z$. Let $L$ be the projection of $O$ on $d$. Prove that $(ALX), (BLY), (CLZ)$ are coaxal.

Problem 3:  Given triangle $ABC$, its circumcircle $(O)$ and a vector $\vec{u}$ . Let $d$ be a line such that $d // \vec{u}$. $d$ intersects $AB, AC$ at $Y, Z$. Let $M, N$ be the midpoints of  $BY, CZ$, respectively; $L$ be the projection of $O$ on $d$. Prove that when $d$ moves, the line through $L$ and perpendicular to $MN$ passes through a fixed point.

Have fun! 🙂

## From a problem about concyclic in Mathematics and Youth magazine

Problem 1 (M&Y):  Given 4 points $A_1, A_2, A_3, A_4$ in the plane and an arbitrary point $P$ such that $P$ is not lie on any circumcircle of these points.
1. Prove that the pedal circles of $P$ wrt triangles $A_1A_2A_3, A_2A_3A_4, A_1A_3A_4, A_1A_2A_4$ are concurrent.

We will not discuss about the solution of this problem. It can be solved by using direct angle. Now from problem 1, I found 3 problem involved as below:

Problem 2:  Given 5 points $A_1, A_2, A_3, A_4, A_5$ in the plane and an arbitrary point $P$ such that $P$ is not lie on any circumcircle of these points.
1. Prove that the pedal circles of $P$ wrt triangles $A_1A_2A_3, A_2A_3A_4, A_1A_3A_4, A_1A_2A_4$ concur at a point $X_5$.
2. Similarly we define $X_1, X_2, X_3, X_4$. Prove that $X_1, X_2, X_3, X_4, X_5$ are concyclic.

Problem 3:  Given a cyclic quadrilateral $ABCD$. Let $X$ be an arbitrary point in the plane such that $X$ is not lie on $(ABCD)$. Prove that the centers of the pedal circles of $X$ wrt $ABC, BCD, CDA, DAB$ are concyclic.

Problem 4:  From 4 non-collinear points $A_1, A_2, A_3, A_4$, we define the point $A_n$ is the orthocenter of the triangle $A_iA_jA_k (1\leq i. Then we have a set $\{A_1, A_2, A_3,..., A_n\}$. Prove that the pedal circle of any point $A_x$ wrt triangle $A_iA_jA_k (1\leq i and the 9-point circles of all the triangle $A_iA_jA_k$ are concurrent.

3 months ago, I wrote the solutions for these problems in an article but what a pity, I lost it. So I will leave links to these problems:

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=422801

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=422802

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=422803

Interestingly, the last problem of this topic is the corollary of problem 2.