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to the cross product 
reveals that a negative determinant [of 
] corresponds to a matrix which reverses the direction of the cross product of two vectors.[/quote]
and 
, but am confused on how to prove this statement for general vectors 
and 
although its supposed to be a quick comparison.
chance of landing heads when flipped. Krithik is playing a game where he starts with 
point. Every turn, he flips the coin, and if it lands heads, he gains point, and if it lands tails, he loses point. However, after the turn, if he has a negative number of points, his point counter resets to . Krithik wins when he earns 
points. Find the expected number of turns until Krithik wins.
be an isosceles triangle with 
Let 
be a point on the circumcircle of on minor arc 
Let 
intersect the extension of 
at 
Let 
be the midpoint of segment 
and let 
be the intersection of 
and 
Let the extension of 
intersect 
and the circumcircle of at 
and 
respectively. Given that 
and 
compute the length of 
is equal to the sum of the areas of areas 
and 
.
and 
and 
be the feet of the altitudes to 
from points 
and 
respectively. Also, extend sides 
and 
(we will deal with the case where 
is a rectangle later).
,
,
and 
.
,![$[EAF] + [EDC] = [BFD]$](http://latex.artofproblemsolving.com/f/4/a/f4afed0aad7da769bb1ecd041cd71cb9760b0084.png)
.
.
.
Now, we only need to prove that last equation is true to prove that the different areas are equal.
.
First, cross multiplying 
and solving for 
,
Next, cross multiplying 
and solving for 
Now, we can set those two equations equal, cross multiply, and simplify:
so we see that the area sums 
,