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Pascal theorem

Pascal theorem

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Problem is proposed by Dao Thanh Oai.

Brianchon generalization

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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.

Mixtilinear2Proof: See here

 

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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 thenJ \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.

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Circumscribed quadrilateral

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.


Link to problem above: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=355881

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On the figure of IMO 2009 P2 Generalization

For the generalization, see the file below:

IMO P2 generalization

We found some nice properties about this figure.

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! 🙂

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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<j<k\leq n-1). 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<j<k\leq n, x\neq i, j, k) 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.

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