Dear Clark Kimberling, ... I did some explorations on the problem, which may be of some help, but I don't have a complete synthetic proof at the moment. A...
7368
Aad Goddijn
aadgoddijn
Jul 16, 2003 10:06 pm
Dear Nicolas (and Clark) Sorry Nikolas; I did not see your remarks before sending my humble observations. I observed the same as Rene de Vogelaere proved. ...
7369
Vijaya Prasad Nalluri
vprasad_nalluri
Jul 18, 2003 4:04 am
Dear Hyacinthos members, The following problem was set for the 23rd Int. Math. Olympiad: A non-isosceles triangle ABC is given with sides a,b,c. A',B',C' are...
7370
Darij Grinberg
darij_grinberg
Jul 18, 2003 5:31 am
Dear Vijaya Prasad Nalluri, ... The problem is relatively simple if looked from the right point. At first, the points D', E' and F' lie on the circumcircle of...
7371
Alexey.A.Zaslavsky
zasl@...
Jul 18, 2003 5:31 am
Dear Vijaya Prasad Nalluri! The lines AD', BE', CF' concur in point G' isogonally conjugated to Gergonne point G. Alexey Myakishev proved that G' is a...
7372
Jean-Louis Ayme
jeanlouisayme
Jul 18, 2003 7:37 am
Dear Hyacintists the book intitled "Méthodes et Techniques en Géométrie, A propos de la droite de Newton" is available by Editions Ellipses (Paris) beyond...
7373
Nikolaos Dergiades
ndergiades
Jul 18, 2003 8:02 am
Dear Aad, ... figure, ... It was also a problem for me. But Rene de Vogelaere and Paul Yiu (who sent me yesterday an unfinished note on Rabinowitz points) ...
7374
Paul Yiu
yiuatfauedu
Jul 18, 2003 1:53 pm
Dear Vijaya, Darij and Alex, [DG]: ... ... What a coincide. Yesterday I was looking at the same problem getting almost the same conclusions. I also look at the...
7375
Darij Grinberg
darij_grinberg
Jul 18, 2003 2:23 pm
Four lines a, b, c, d meet at the points A = a /\ c; B = b /\ c; A' = a /\ d; B' = b /\ d; C = c /\ d; D = a /\ b. The circumcircles of...
7376
Darij Grinberg
darij_grinberg
Jul 18, 2003 4:24 pm
Dear Paul Yiu and Alexey Zaslavsky, Now I understand your reasoning: Since D' is the reflection of D in the perpendicular bisector of EF, but this...
7377
Nikolaos Dergiades
ndergiades
Jul 18, 2003 9:31 pm
Dear Darij, ... From the similarity of triangles QAB, QA'B' because ang QAC = ang QA'C and ang QBC = ang QB'C we conclude that ang QPC = ang QP'C and hence the...
7378
Darij Grinberg
darij_grinberg
Jul 20, 2003 6:26 am
Dear Nikolaos Dergiades, ... Aha... this is because P and P' are corresponding points in the triangles QAB and QA'B'. ... MANY THANKS!! Let me explain why I...
7379
Darij Grinberg
darij_grinberg
Jul 20, 2003 6:33 pm
I have found a nice conjecture: Given a triangle ABC with circumcenter O, any line l through O intersects the circles OBC, OCA, OAB at X, Y, Z, then the lines ...
7380
Darij Grinberg
darij_grinberg
Jul 20, 2003 6:33 pm
In the Crux Mathematicorum #1579 problem, S. Kotani and H. Fukagawa asked to show that if similar rectangles ABDE, CAFG, BCHI are erected on the sides of a ...
7381
Nikolaos Dergiades
ndergiades
Jul 21, 2003 4:21 pm
Dear Darij, You wrote: Here is the statement of the Droz-Farny theorem: Let ABC be a triangle with the orthocenter H and let two lines g1, g2 pass through the ...
7382
Nikolaos Dergiades
ndergiades
Jul 21, 2003 10:22 pm
Dear Darij, your conjecture ... ********** is true. LEMMA1 The isogonal tranformation of the circle OBC is the circle HBC. LEMMA2 If P is a point on the circle...
7383
nickreingold
Jul 22, 2003 1:25 pm
Dear Darij Grinberg, ... Here is another way to establish (c). In addition to the above notation, let X3, Y3, and Z3 be the midpoints of X1X2, Y1Y2, and Z1Z2 ...
7384
Darij Grinberg
darij_grinberg
Jul 23, 2003 9:49 am
Dear Nick Reingold, Thank you very much for the nice projective proof! I am just going to fill in some steps of the proof you have not mentioned directly to...
7385
Darij Grinberg
darij_grinberg
Jul 23, 2003 10:07 am
In Hyacinthos message #7120, Alexey Zaslavsky has proposed a very nice theorem which I restate as follows: Let A'B'C' be the reflection of a triangle ABC ...
7386
Darij Grinberg
darij_grinberg
Jul 23, 2003 11:21 am
Dear Alexey Zaslavsky, ... This refers to your ... Could you please elaborate more? I don't understand this proof. To which hexagon do you apply Brianchon's ...
7387
Darij Grinberg
darij_grinberg
Jul 23, 2003 7:17 pm
Dear Alexey Zaslavsky, ... Now I have understood it myself. Brianchon's theorem is applied to the hexagon A0B'A'B0XY, where A0 and B0 are the infinite points...
7388
mustafa yagci
yagcimustafa
Jul 24, 2003 2:19 pm
ABC is a triangle and D is a point on [BC]. The circumcircle of the triangle ADC intersects AB at K other than A. Let the perpendicular bisector of [DC]...
7389
Nikolaos Dergiades
ndergiades
Jul 24, 2003 3:33 pm
... circumcircle of the triangle ADC intersects AB at K other than A. Let the perpendicular bisector of [DC] intersect AC at F and DC at E. If AF/FC=CE/EB then...
7390
dick tahta
dick@...
Jul 25, 2003 8:57 am
I enjoyed Nikolaos' neat solution of the problem. Meanhwhile I have been unable to manipulate the corresponding (barycentric) algebra and wonder whether...
7396
- \&...
seidovzf
Jul 28, 2003 12:14 pm
Dear all, thank you who helped or wished to help with info, as usually, any reference gives references to previous sources of references (* e.g. ... *)... ...
7402
Darij Grinberg
darij_grinberg
Jul 31, 2003 3:18 pm
In Hyacinthos message #7369 and later, we discussed a triangle ABC, the points D, E, F where its incircle touches the sides, and the reflections D', E', F' of...
7403
yagcimustafa
Jul 31, 2003 5:21 pm
ABCD is a convex quadrilateral with area 18. If |AB|+|BD|+|DC|=12 then |AC|=? This problem was asked in a math olympiad elimination in Turkey.I couldn't solve...
7404
Paul Yiu
yiuatfauedu
Jul 31, 2003 5:51 pm
Dear Darij, Both of your conjectures are correct. Best regards Sincerely Paul...
7406
Nikolaos Dergiades
ndergiades
Jul 31, 2003 7:24 pm
Dear Paul, ... circumcircle and ... that X is ... segment AP ... ******* Sorry if I am not right, but I do not remember an answer to this message. Triangle ABC...
7407
Nikolaos Dergiades
ndergiades
Aug 1, 2003 6:12 am
Dear Mustafa your problem ... is with other words the 1st problem of the 18 IMO (1976). If a = AB, b = BD, c = DC, a+b+c=12 x = angABD, y = angBDC then ...
