# Geometry 几何

## theorems 定律

1)In any triangle $ABC$, $AB\lt AC+BC$.
1)对于任意三角形 $ABC$$AB\lt AC+BC$
2)For any right-angled triangle $ABC$ with side $a$ as hypotenuse, $a^2=b^2+c^2$.
2)对于任意以 $a$ 为斜边的直角三角形 $ABC$$a^2=b^2+c^2$
3)For any triangle $ABC$, $\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$.
3)对于任意三角形 $ABC$$\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$
4)For any triangle $ABC$, $a^2=b^2+c^2-2bc\cos A$.
4)对于任意三角形 $ABC$$a^2=b^2+c^2-2bc\cos A$

## proof 证明

Please try to prove 2) and 3) yourself using size relationships.

# Trigonometry 三角学

## theorems 定律

1) $\sin ^2 \theta + \cos ^2 \theta = 1 {\kern 5em} ( \sin ^2 \theta = ( \sin \theta )^2)$
2) $\sin (A\pm B) =\sin A \cos B \pm \cos A\sin B$
3) $\cos (A\pm B)=\cos A\cos B \mp \sin A\sin B$
4) $\tan (A\pm B)=\dfrac{\tan A \pm \tan B}{1\mp \tan A\tan B}$
5) $\sin 2\alpha =2\sin \alpha \cos \alpha$
6) $\cos 2\alpha = \cos ^2\alpha -\sin^2\alpha =2\cos ^2\alpha -1=1-2\sin ^2 \alpha$
7) $\tan 2\alpha = \dfrac{2\tan \alpha}{1-\tan ^2\alpha}$
8) $\sin ^2 A=\dfrac{1-\cos 2A}{2}$
9) $\cos ^2 A=\dfrac{1+\cos 2A}{2}$
10) $\tan ^2 A=\dfrac{1-\cos 2A}{1+\cos 2A}$
11) For a unit circle on a coordinate grid with its center at the origin, point $P(\cos \theta ,\sin \theta )$ is on the circle, and line $\overline{OP}$ and the x-axis form an angle of $\theta$.
11)对于一个表示在坐标轴上且圆心在原点的单位圆，点 $P(\cos \theta ,\sin \theta )$ 一定在该圆上，而且直线 $\overline{OP}$ 和x-轴的夹角为 $\theta$
12) $\tan (90^\circ -A)=\cot (A)$

## proof 证明

11) will be used but not proved 11) 会被用来证明，但它本身不会被证明
Try to prove 1),5),6),7).

Other proofs 其他证明:

Note by Jeff Giff
2 months, 3 weeks ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

LOL Your first(second) line is wrong

- 2 months, 2 weeks ago

P.S. I added one proof to the chain

- 2 months, 2 weeks ago

In China the correct order is 2,3,1,5,6,7??? lol

- 2 months, 2 weeks ago

LOL

- 2 months, 2 weeks ago

Umm... actually there are two parts, and the numbers refer to the theorem in the part :)

- 2 months, 2 weeks ago

oops... third

- 2 months, 2 weeks ago

Where?

- 2 months, 2 weeks ago

ABC, AB\gt AC+BC

- 2 months, 2 weeks ago

Wait...whoa! Thanks!

- 2 months, 2 weeks ago

or fourth with the title

- 2 months, 2 weeks ago

@Páll Márton, I’ve been thinking of a proof for 8,9,10, but I have no idea. Can you help me?

- 2 months, 2 weeks ago

- 2 months, 2 weeks ago

Never mind :) I finally found the proof online

- 2 months, 2 weeks ago