Trig Ceva, for hexagons this time
by math_explorer, Sep 25, 2010, 6:33 AM
I've given up putting all the 
signs in every single sine expression, at least for this post.
Lemma 1. If point
is inside a polygon $V_1V_2\ldotsV_n$, the product
![\[ \frac{\sin V_1V_2O \sin V_2V_3O \ldots \sin V_nV_1O}{\sin V_2V_1O \sin V_3V_2O \ldots \sin V_1V_nO} = 1 \]](//latex.artofproblemsolving.com/d/f/9/df9e2a2b1d56a5b7a19ed8d72354d41f2f832b32.png)
Proof:
. Take the product.
Meh, I don't feel like finding a problem now. I've been putting off a lot of stuff.
signs in every single sine expression, at least for this post.Lemma 1. If point
is inside a polygon $V_1V_2\ldotsV_n$, the product![\[ \frac{\sin V_1V_2O \sin V_2V_3O \ldots \sin V_nV_1O}{\sin V_2V_1O \sin V_3V_2O \ldots \sin V_1V_nO} = 1 \]](http://latex.artofproblemsolving.com/d/f/9/df9e2a2b1d56a5b7a19ed8d72354d41f2f832b32.png)
Proof:
. Take the product.April wrote:
Let 
and 
be two points in interior of the given triangle 
. Let 
, 
, 
be the traces on sides 
, 
, 
, respectively with the lines through and parallel to 
, 
, 
. Let 
, 
, 
be the traces on sides , , , respectively with the lines through and parallel to 
, 
, 
.
Prove that
, 
, 
are concurrent if and only if 
, 
, 
are concurrent.
and 
be two points in interior of the given triangle 
. Let 
, 
, 
be the traces on sides 
, 
, 
, respectively with the lines through and parallel to 
, 
, 
. Let 
, 
, 
be the traces on sides , , , respectively with the lines through and parallel to 
, 
, 
.Prove that
, 
, 
are concurrent if and only if 
, 
, 
are concurrent.
and 
.
,
,
.
,
,
.
,
,
are concurrent iff 
,
,
are concurrent.
,
,
are just the lines 
,
,
,
,
,
.
(Law of Sines bash), and 
.
and 
,
.
,
.Meh, I don't feel like finding a problem now. I've been putting off a lot of stuff.
May 2020
January 2020