Dear Luis, Catalin, Paul and Nikolaos! I can propose a similar problem. Let G_a, G_b, G_c be the touching points of the incircle with BC, CA, AB. Then G_aG_b,...
20099
Kafka Catalin
kafka_mate
Jun 20, 2011 7:21 am
Dear Alexey,  I attach a proof for your problem. I hope that is a good proof for you.  Best regards, Catalin Barbu     ... From: Alexey Zaslavsky...
20100
Barry Wolk
wolkbarry
Jun 20, 2011 9:45 pm
... Here is another elementary solution. Let the incircle meet BC at X. AP = AHa - PHa AHa = rs/(a/2) = r(a+b+c)/a PHa / r = PHa / IX = HaMa / XMa Wlog,...
20101
Nikolaos Dergiades
ndergiades
Jun 20, 2011 11:12 pm
Dear Alexey, your problem ... can be stated as: If the diameter of ABC from A meets BC at M then M lies on the A' median of the tangential trianle A'B'C' of...
20102
cbanerjee.2011
Jun 21, 2011 4:40 am
In a triangle A1B1C1, reflect the vertices on the opposite sides to get A2B2C2. Do the same thing in the new triangle. Prove that after infinite steps the...
20103
Angel
amontes1949
Jun 23, 2011 10:20 am
Dear Hyacinthists Let D = AH /\ IM_a, E = BH /\ IM_b, and F = CH /\ IM_c. (AD = BE = CE = r) Let A' be the reflection of D in A, B' the reflection of E in B,...
20104
Alexey Zaslavsky
zasl@...
Jun 23, 2011 10:58 am
Dear colleagues! Let the incircle of triangle ABC touche its sides in A', B', C'; I and G are the incenter and the Gergonne point. Then the hyperbolas ABCIG...
20105
Antreas Hatzipolakis
xpolakis
Jun 23, 2011 11:03 am
Let ABC be a triangle with A = 90 d. Let D be a point in AC such that inradius of BDA = inradius of BDC := d Find all triangles ABC with a,b,c,d integers. ...
20106
Sung Hyun Lim
bbblow
Jun 24, 2011 4:05 am
Hello Alexey, I found a generalization to your problem, and a generalized property. If you take A'B'C' as the reference triangle instead, then I becomes ...
20107
Antreas Hatzipolakis
xpolakis
Jun 24, 2011 5:36 am
Correction: Let ABC be a triangle with A = 90 d. Let D be a point in AC such that inradius of BDA = inradius of BDC := d Find all triangles ABC with a,b,c,d...
20108
Francisco Javier
garciacapitan
Jun 24, 2011 5:55 am
Dear Alexey and Sung Hyun, I calculated these centers of homothety: Let H1 the hyperbola ABCIG H2 the hyperbola A'B'C'IG Z1 the center of H1 Z2 the center of...
20109
Antreas Hatzipolakis
xpolakis
Jun 24, 2011 10:57 am
Solution HERE <http://anthrakitis.blogspot.com/2011/06/inradius-5.html> (hope there are no computational errors!) APH On Thu, Jun 23, 2011 at 2:02 PM, Antreas...
20110
Francisco Javier
garciacapitan
Jun 24, 2011 2:19 pm
Dear Alexey and Sung Hyun: According to Sung Hyun's nice generalization we have: Let X a point Y barycentric product of X and (anticomplement of X) A'B'C'...
20111
Francisco Javier
garciacapitan
Jun 24, 2011 2:22 pm
Sorry, I meant A'B'C' anticevian triangle of Y Best regards, Francisco Javier....
20112
Antreas Hatzipolakis
xpolakis
Jun 25, 2011 2:55 pm
Let ABC be a triangle and P a point. If B,C,P are fixed, which is the locus of A such that the NPC of ABC passes through P? The center N of the NPC is lying on...
20113
Francisco Javier
garciacapitan
Jun 25, 2011 3:50 pm
Only for the first case: If B=(-a,0), C=(a,0) and P=(u,v) then A lies on the conic v (x^2 - y^2) - 2 u x y + 2 (u^2 + v^2) y - a^2 v = 0....
20114
Francisco Javier
garciacapitan
Jun 25, 2011 5:26 pm
This is the rectangular hyperbola centered at P through B and C. This hyperbola can be constructed as follows (See Eagles' Constructive Geometry of Plane...
20115
Barry Wolk
wolkbarry
Jun 25, 2011 7:55 pm
[Angel] ... This generalizes. For any real f, define A' B' C' by the vector equations vec(A,A') = f * vec(A,D), vec(B,B') = f * vec(B,E), vec(C,C') = f *...
20116
Francois Rideau
francoisride...
Jun 26, 2011 5:44 pm
There is a special case where your theorem is false! This is the case when at the first step, points A2B2C2 are on a same line! In this case d(X(3), X(4)) = 2R...
20117
Chandan Banerjee
cbanerjee.2011
Jun 26, 2011 7:44 pm
Ya. You are correct. I am sorry. I forgot to mention that the initial triangle is acute-angled triangle....
20118
Francois Rideau
francoisride...
Jun 26, 2011 10:30 pm
If we look at the affine map f:A1B1C1 --> A2B2C2, then the trace of the linear map of f is 4R and its determinant 4R² -OH². That's why points A2, B2, C2 are...
20119
Alexey Zaslavsky
zasl@...
Jun 27, 2011 6:37 am
Dear Francisco and Sung Hyun! This result can be generalised as next projective theorem. Let triangles ABC and A'B'C' be polar wrt conic k and P be their...
20120
Francois Rideau
francoisride...
Jun 27, 2011 12:20 pm
I correct some typos: you mustv read: If we look at the affine map f:A1B1C1 --> A2B2C2, then the trace of the linear part of f is -4R and its determinant 4R²...
20121
Francois Rideau
francoisride...
Jun 27, 2011 12:44 pm
Dear Banerjee In fact, your process may converge even in case the first triangle is not acute but in this case the situation is very complicated to handle. ...
20122
Chandan Banerjee
cbanerjee.2011
Jun 27, 2011 6:32 pm
Dear Francois, I know that it will converge even in case when the triangle is not acute, but as u said there are some special cases when the triangle will...
20123
Sung Hyun Lim
bbblow
Jun 29, 2011 9:13 am
Dear everyone, Does anyone know a simple proof of the statement: <Two triangles share a circumconic iff they share an inconic>? Sung Hyun [Non-text portions of...
20124
Alexey Zaslavsky
zasl@...
Jun 29, 2011 9:57 am
Dear Sung Hyun! Does anyone know a simple proof of the statement: <Two triangles share a circumconic iff they share an inconic>? This follows from Poncelet...
20125
Forum Geometricorum
ForumGeom@...
Jun 29, 2011 1:49 pm
The following paper has been published in Forum Geometricorum. It can be viewed at http://forumgeom.fau.edu/FG2011volume11/FG201114index.html The editors Forum...
20126
Jean-Louis Ayme
jeanlouisayme
Jun 30, 2011 11:27 am
Dear Hyacinthists, is there some geometers who have an article concerning the proof of Salmon's points in relation with the Pascal's theorem? Thanks in advance...
20127
Ricardo Barroso
ricardobca
Jul 1, 2011 5:07 am
Amigos de Hyacinthos: Os deseo buen verano. He preparado un extra con un problema de un autor español del XIX, Zoel García de Galdeano, navarro, y un...
