In $\triangle ABC$ , $c^2 = a^2 + b^2 - 2ab \cos C$ahmedittihad wrote:What is cosine law?
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- Sun Apr 09, 2017 7:59 pm
- Forum: Junior Level
- Topic: geomerty
- Replies: 10
- Views: 8371
Re: geomerty
- Sat Apr 08, 2017 11:32 pm
- Forum: Junior Level
- Topic: geomerty
- Replies: 10
- Views: 8371
Re: geomerty
The simplest solution seems to me:
1. Use Cosine Law to find $AB$
2. Angle Bisector Theorem to find $AD$
3. Cosine Law on $\triangle ADC$ to find $CD$
1. Use Cosine Law to find $AB$
2. Angle Bisector Theorem to find $AD$
3. Cosine Law on $\triangle ADC$ to find $CD$
- Fri Apr 07, 2017 9:34 pm
- Forum: Junior Level
- Topic: Beginner's Marathon
- Replies: 68
- Views: 44733
Re: Beginner's Marathon
$\text{Problem }9$ There is a billiard ball rolling on a circular table. Everytime it hits the edge, it gets reflected (Assume the ball hits the circle at X . The course of the ball gets reflected with respect to the tangent from X). Prove that, if the ball goes through a point $P$ three times, the...
- Thu Apr 06, 2017 10:55 pm
- Forum: Algebra
- Topic: Find largest pos int n with special condition
- Replies: 1
- Views: 4987
Find largest pos int n with special condition
Find the largest possible positive integer $n$, such that there exists $n$ distinct positive real numbers $x_1, x_2, \cdots, x_n $ satisfying the following inequality:
for any $ 1 \le i, j \le n$
$ (3x_i - x_j) (x_i - 3x_j) \ge (1 - x_ix_j) ^2$
for any $ 1 \le i, j \le n$
$ (3x_i - x_j) (x_i - 3x_j) \ge (1 - x_ix_j) ^2$
- Sun Apr 02, 2017 8:39 pm
- Forum: Secondary Level
- Topic: Find the angle
- Replies: 2
- Views: 3046
Re: Find the angle
I solved it by Trig-Ceva. I don't wanna write those trigging right now. If anyone solved synthetic, please post ur solu.
- Sun Apr 02, 2017 8:31 pm
- Forum: Geometry
- Topic: All Russian Math Olympiad 2010
- Replies: 2
- Views: 3723
Re: All Russian Math Olympiad 2010
Well, I first missed the point that segments $BC$ and $XY$ intersects at $M$. After noticing this the problem is easy. My solution is similar to Ittihad's. WLOG $AB>AC$. Take the $A$-excircle of $\triangle ABC$. it touches $BC$ at $D$ and the extensions of $AB$ and $AC$ at $E$ and $F$ respectively. ...
- Fri Mar 31, 2017 9:49 pm
- Forum: Geometry
- Topic: CGMO 2012/5
- Replies: 3
- Views: 3903
Re: CGMO 2012/5
It's a well known fact that $A,I,O$ are collinear. $AD=AE$ along with $\angle DAO = \angle EAO $ implies that triangle $DAO$ and $EAO$ are congruent. The rest is trivial.
- Fri Mar 31, 2017 9:33 pm
- Forum: Geometry
- Topic: Geometric Ineq
- Replies: 1
- Views: 2548
Geometric Ineq
Let $h_a$, $h_b$, $h_c$ be the lengths of the altitudes from $A$, $B$, $C$ respectively of a triangle $ABC$. Let $P$ be any point inside the triangle. Prove that
$\frac{PA}{h_b+h_c} + \frac{PB}{h_c+h_a} + \frac{PC}{h_a+h_b} \ge 1$
$\frac{PA}{h_b+h_c} + \frac{PB}{h_c+h_a} + \frac{PC}{h_a+h_b} \ge 1$
- Mon Mar 27, 2017 6:39 pm
- Forum: Social Lounge
- Topic: BDMO Forum Mafia #1
- Replies: 52
- Views: 57811
Re: BDMO Forum Mafia #1
I'll play. But I wanna know, how the cards will be distributed?
- Sat Mar 25, 2017 8:20 pm
- Forum: Social Lounge
- Topic: BDMO Forum Mafia
- Replies: 5
- Views: 4480
Re: BDMO Forum Mafia
খেলা হবে। ✌✌✌