Geometry Dash
by djmathman, Aug 13, 2015, 1:15 AM
Being removed from official math contests, I now essentially have the freedom to practice math whenever I want without the pressure of competition. (Technically, I've always studied math without an organized study regimen, but the extra freedom is a new plus.) As a result, I'm able to practice math a little... unconventionally.
One of these new ways came through v_Enhance's new GeoGuessr handout here. I've never really explored the mixtillinear incircle configuration before to that extent, so I tried an experiment: my goal was to see how many of the listed facts I could prove/recall the proof to within the span of an hour. It was actually quite a fun exercise!
Just for kicks, here are my mostly-unedited sketches of my proofs. Edits are denoted in parentheses.
Also triple speed ship in Electrodynamix is so hard ahhh
One of these new ways came through v_Enhance's new GeoGuessr handout here. I've never really explored the mixtillinear incircle configuration before to that extent, so I tried an experiment: my goal was to see how many of the listed facts I could prove/recall the proof to within the span of an hour. It was actually quite a fun exercise!
Just for kicks, here are my mostly-unedited sketches of my proofs. Edits are denoted in parentheses.
,
)
to the tangent to 
parallel to 
.
is taken to 
yields 
,
collinear, and 
angle bis. 
midpt.
be this top point. Note that 
(easiest way to prove this is to note both lines are 
to 
)
,
is a 
-
passes through the midpoint of 
.
,
homothetic, so 
,
.
follows by homothety wrt 
.
-
,
.
and 
is refl. of 
,
passes through refl. of 
.
,
.
and 
so 
.
(both isoc. and have vert. angles 
)
.
cyclic. Similar reasoning applies for 
.
.
is tangent to 
.
and 
.
where 
.
is cyclic as desired.
,
to 
and 
to itself. This I could not do in the allotted hour.
)
and 
.
and 
,
is cyclic. Thus 
.![\[\angle C_1Y_1C=\angle(M_CM_B, BC)=\frac12(\widehat{M_BC}-\widehat{M_CB})=\frac12\left(\widehat{M_BA}-\widehat{M_BC}\right)=\dfrac12\widehat{XA}\]](http://latex.artofproblemsolving.com/1/7/8/178516b2d4b33126e9fe4949e93b8522b763ad9a.png)
as well. Thus 
,
.
passes through the top point, and 
,
also passes through the bottom point. QEDAlso triple speed ship in Electrodynamix is so hard ahhh
This post has been edited 2 times. Last edited by djmathman, Aug 13, 2015, 1:17 AM
Reason: fixed typo
Reason: fixed typo
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