PUMaC and some Geo
by djmathman, Nov 18, 2013, 4:50 AM
PUMaC 2013 was actually a lot of fun! In terms of the math itself, I got pretty wrecked; got #2,3 on Geo A and I think #1-5 in NT A. (The first several questions in NT weren't bashy/didn't require Olympiad level ideas, and that's probably why I got so many. On the contrary, if I got #1-5 in Geo I might have had a better shot of placing since the test seemed so much harder.) Also, our team unintentionally won the Puzzle Hunt, so yay!
I also got to meet a bunch of new AoPSers, so I guess it's time to update my bandwagon again, with bold usernames representing changes:
And speaking of Princeton, one geometry boss who resides in Princeton goes by Cosmin, and speaking of Cosmin, I've been relooking through his Geo 3 notes and trying some of the problems.
I also got to meet a bunch of new AoPSers, so I guess it's time to update my bandwagon again, with bold usernames representing changes:
And speaking of Princeton, one geometry boss who resides in Princeton goes by Cosmin, and speaking of Cosmin, I've been relooking through his Geo 3 notes and trying some of the problems.
Quote:
Let 
be a triangle and let 
, 
be points on the sides 
. Let 
be the incircle of and let 
be the tangency points of with the same sides 
and 
respectively. Furthermore, draw the tangents from and to which are different from the sidelines of and take the tangency points with to be 
and 
, respectively. Prove that the lines 
and 
are concurrent.
be a triangle and let 
, 
be points on the sides 
. Let 
be the incircle of and let 
be the tangency points of with the same sides 
and 
respectively. Furthermore, draw the tangents from and to which are different from the sidelines of and take the tangency points with to be 
and 
, respectively. Prove that the lines 
and 
are concurrent.
,
,
,
are collinear. Similarly, by Pascal on 
,
,
,
are collinear. Therefore 
and we are done. 
Quote:
In triangle with 
, the point 
denotes the orthocenter. The points 
and are the foot of the perpendiculars from 
and 
, respectively. The point 
is symmetric of 
with respect to . If 
, 
, and 
, prove that the lines 
, 
, and 
are concurrent.
with 
, the point 
denotes the orthocenter. The points 
and are the foot of the perpendiculars from 
and 
, respectively. The point 
is symmetric of 
with respect to . If 
, 
, and 
, prove that the lines 
, 
, and 
are concurrent.
;
are collinear, and since 
(based off symmetry about 
)
as well. Note that 
,
is isosceles. Thus 
,
is cyclic. This, combined with the well-known fact that 
is cyclic, gives 
,
is also cyclic and 
.
is the orthocenter of 
,
,
April 2025
November 2024