... Show ... Let BH,IF intersect at X; CH,IE intersect at Y The figure IXHY enclosed by parallels BH,IE and CH,IF is a parallelogram So HI bisects XY...(i) ...
Dear Darij, in message 9725 you wrote about the antigonal conjugate: ... ... I think it is: The antigonal conjugate D' of D is the isogonal conjugate of the...
Dear Eric, ... Oh yes, sorry. Thanks for the remark. By the way, the proof is easy: If D1 is the isogonal conjugate of D, then we can readily see that < BD1C =...
... Suppose that your line L intersects BC at W. Your locus is a cubic through B,C,W; the three asymptots are the bisectors of (BC, L) and the perpendicular to...
Dear Antreas, Let H_b (resp. H_c) be the pencil of couple of perpendicular lines (degenerate rectangular hyperbolas) through B (resp. C). Let D_b (resp D_c)...
As some of you know, I participate in the German IMO training; this means I am one of the 16 students which are being prepared for a possible participation in...
Dear Darij and Barry! ... This problem can be reformulated. Let given two circles with centers O1, O2 and their two common tangents: external and internal. l1,...
Alexey.A.Zaslavsky
zasl@...
May 7, 2004 9:03 am
9771
I see two very nice problems in the St. Petersburg Mathematics Olympiad 2002, Elimination Round: Problem 10/6 in an equivalent version ... The tangent to the...
... I've only thought about 11/2, which seems to be the easiest one of these 3. By a quick angle chase we can show that the circle (IXY) passes through the S,...
Dear Grobber, In Hyacinthos message #9772, you wrote: [on 11/2] ... This is nice! Here is the proposed solution: The triangles AXI and AYI are congruent, since...
Dear Darij, ... ..... Here is the proposed solution: The triangles AXI and AYI are congruent, since AX = AY, AI = AI and < XAI = < YAI. Hence, XI = YI, and...
Dear Alexey and Darij, In msg 9770 Alexey wrote: ... This problem can be reformulated. Let given two circles with centers O1, O2 and their two common tangents:...
Dear Darij! -> ... This is the partial case of next fact. Let given the triangle ABC and the point P. A', B', C' are the second common points of AP, BP, CP and...
Alexey.A.Zaslavsky
zasl@...
May 11, 2004 12:08 pm
9777
Dear Darij! ... After polar transformation we receive next proposition. Let I is the incenter of ABC, the line passing through I and parallel to AA' intersect...
Alexey.A.Zaslavsky
zasl@...
May 11, 2004 1:03 pm
9778
To generate a road in ray tracing, I need to get a section of cylinder when I just know the road width and the height in the midle (the edges are at 0). More...
You wrote, on your site, by trigonometric evaluation : W = 2R sin(a) H = R [ 1 - cos(a) ] hence : R cos(a) = R - H In fact, you have rectangular triangle with...
The following paper has been published in FG. It can be viewed at http://forumgeom.fau.edu/FG2004volume4/FG200409index.html The editors Forum Geometricorum ...
ForumGeom
ForumGeom@...
May 12, 2004 2:17 pm
9783
Dear flatt ? I could not access your site via the link you gave, but it doesn't matter as the follow up message from Gilles Boutte gave the essential...
for your help. The solution is implemented and works fine. It is in fact finding the radius of the circle going through 3 points. It will be used for road and...
The following problem in Euclidean geometry was posted in other places by Gerry Wildenberg <gwildenb@...> but to no avail. I offered to forward...
Ken Pledger
Ken.Pledger@...
May 13, 2004 2:58 am
9786
Dear all, With triangle ABC let D be the point in the plane of ABC such that AB^2 + CD^2 = AC^2 + BD^2 = AD^2 + BC^2. Which point is D? Kind regards, Floor....
Dear Floor! ... As AD^2-BD^2=AC^2-BC^2, CD is perpendicular to AB. So D is the orthocenter. Sincerely Alexey...
Alexey.A.Zaslavsky
zasl@...
May 13, 2004 9:28 am
9788
Dear Floor ... The locus of M such as BM^2-CM^2 = BA^2-CA^2 is clearly the A- altitude. Thus D is (very) probably the orthocenter. Friendly. Jean-Pierre...
Dear Hyacinthians, The following theorem is an example of a non-standard triangle geometry result because the proof (at least, the proof I know) uses a ...
In Hyacinthos message #3957, Paul showed an interesting fact about circumcevian triangles originating in a question of Antreas. I will restate this fact with...
My generalization of Chaogold's result in http://mathlinks.ro/viewtopic.php?t=5058 Let P be a point in the plane of a triangle ABC; an arbitrary conic k...
... Precisely, the three pairwise radical axis of the 3 circles , i.e. the sidelines of ABC, are concurrant, this fact is impossible for ABC non-degenerate. Y,...
