-
this guy is an absolute legend. much love wherever you are Michael
by
LeonidasTheConquerer, Aug 5, 2024, 9:37 PM
-
amazing blog
by
anurag27826, Jun 17, 2023, 7:20 AM
-
hi i randomly found this
by
purplepenguin2, Mar 1, 2023, 8:43 AM
-
can i be a contributor please?
by
cinnamon_e, Mar 10, 2022, 6:58 PM
-
orzorzorzorzorzorozo
by
samrocksnature, Jul 16, 2021, 8:25 PM
-
2021 post
by
the_mathmagician, May 5, 2021, 3:28 PM
-
Let
be an equilateral triangle of side length 
. Let 
be the point such that 
is the midpoint of 
, and let 
be the incenter of triangle 
. Let 
be the point on line 
such that 
and 
are perpendicular. $ \
by
ARay10, Aug 25, 2020, 5:55 PM
-
Nice blog!
by
User526797, Jan 12, 2020, 4:48 PM
-
oh my gosh it's been so longggggg.... contrib? what does that mean?
by
adiarasel, Dec 1, 2019, 8:31 PM
-
2019 post
by
piphi, Aug 10, 2019, 6:32 AM
-
hi contrib please
by
Emathmaster, Dec 27, 2018, 5:38 PM
-
hihihihihi contrib plzzzzz
by
haha0201, Aug 20, 2018, 3:58 PM
-
contrib please
by
Max0815, Aug 1, 2018, 12:35 AM
-
contrib /charmander
by
mathmaster2000, Apr 16, 2017, 4:59 PM
-
for contrib
by
SomethingNeutral, Mar 30, 2017, 7:57 PM
-
all contrib requests added I usually get to posting every night after running on the treadmill and showering, posting my favorite problem(s) I did during the day.
by
shiningsunnyday, Jun 16, 2016, 4:37 AM
-
Since problems is plural, he might post multiple problems through different posts in one day...
by
zephyrcrush78, Jun 15, 2016, 8:37 PM
-
hi can i have contrib?
by
wu2481632, Jun 15, 2016, 8:36 PM
-
Hello it is called problems of the DAY
by
Temp456, Jun 15, 2016, 8:17 PM
-
10th shout lol
How often will this blog be updated?
by
zephyrcrush78, Jun 15, 2016, 5:24 PM
-
why are we counting shouts
contrib pls
by
MathAwesome123, Jun 15, 2016, 2:55 PM
-
eighth shout
contrib pls
by
skipiano, Jun 15, 2016, 2:21 PM
-
seventh shout
Can I be contrib?
Though probably not because it's not like I'll post anything anyways :/ :/
by
zephyrcrush78, Jun 15, 2016, 1:20 AM
-
sixth shout
new goal: get the shouts that are congruent to 1 mod 5
by
DeathLlama9, Jun 15, 2016, 12:33 AM
-
fifth shout >
by
budu, Jun 14, 2016, 10:04 PM
-
4th yay
by
EpicSkills32, Jun 14, 2016, 9:10 PM
-
third shout
by
skipiano, Jun 14, 2016, 2:35 PM
-
Second shout
by
zephyrcrush78, Jun 14, 2016, 12:06 AM
-
omg yesssss
by
DeathLlama9, Jun 13, 2016, 2:19 PM
by agbdmrbirdyface, Aug 10, 2016, 10:28 PM 
be a circle and let 
be a point outside of 
and 
, respectively. Let 
be the midpoint of segment 
and 
the second intersection of the line 
with circle 
= 
.
be a circle and let 
be a point outside of 
and 
, respectively. Let 
be the midpoint of segment 
and 
the second intersection of the line 
with circle 
= 
.![[asy]
size(200);
defaultpen(fontsize(10pt));
pair A, B, C, P, M, NN;
C = (0,0);
P = (3,0);
path circ = Circle(C, 1);
A = tangent(P, C, 1, 1);
B = tangent(P, C, 1, 2);
draw(circ);
draw(A--P--B--cycle);
M = (A+P)/2;
draw(B--M);
pair [] N = IPs(B--M, circ);
pair [] X = IPs(P--(2*N[1]-P), circ);
draw(P--X[1]);
NN = rotate(180, M)*N[1];
draw(N[1]--A--NN--P, red);
draw(M--NN, red);
draw(P--M--B--P--N[1]--M, green);
dot("$N'$", NN, dir(90));
dot("$A$", A, dir(70));
dot("$B$", B, dir(290));
dot("$C$", C, SW);
dot("$P$", P, dir(0));
dot("$X$", X[1], dir(200));
dot("$M$", M, dir(70));
dot("$N$", N[1], dir(330));
[/asy]](http://latex.artofproblemsolving.com/e/8/b/e8bb7b55cc32a400ba582e30bde425069341e294.png)
,
.
.
.
is harmonic, so lines 
form a harmonic pencil by another well-known lemma. Note that 
intersect 
by construction. Hence, by a well-known lemma about degenerate cases of harmonic divisions, we know that 
.
is a parallelogram since diagonals bisect each other.
as follows:
by cyclic quads and parallel lines, we're done. 
so 
May 2017
April 2017