In $\triangle{ABC}$, $AB = 33, AC=21,$ and $BC=m$ where $m$ is an integer. There exist points $D$ and $E$ on $AB$ and $AC$, respectively, such that $AD=DE=EC=n$ where $n$ is also an integer. Find all the possible values of $m$.
Let quadrilateral $ABCD$ inscribe a circle. If $BE=ED$, prove $$AB^2+BC^2 +CD^2 + DA^2 = 2AC^2$$
In $\triangle{ABC}$, $AE$ and $AF$ trisects $\angle{A}$, $BF$ and $BD$ trisects $\angle{B}$, $CD$ and $CE$ trisects $\angle{C}$. Show that $\triangle{DEF}$ is equilateral.
As shown, $\angle{ACB} = 90^\circ$, $AD=DB$, $DE=DC$, $EM\perp AB$, and $EN\perp CD$. Prove $$MN\cdot AB = AC\cdot CB$$
As shown, in $\triangle{ABC}$, $AB=AC$, $\angle{A} = 20^\circ$, $\angle{ABE} = 30^\circ$, and $\angle{ACD}=20^\circ$. Find the measurement of $\angle{CDE}$.
In tetrahedron $ABCD$, $\angle{ADB} = \angle{BDC} = \angle{CDA} = 60^\circ$, $AD=BD=3$, and $CD=2$. Find the radius of $ABCD$'s circumsphere.
Show that $\sin{x}+2\sin{2x}+\cdots + n\sin{nx}=\frac{(n+1)\sin{nx} - n\sin{(n+1)x}}{2(1-\cos{x})}$
Simplify $\cos{x}\cos{2x}\cdots\cos{2^{n-1}x}$.
Evaluate $\cos\frac{2\pi}{2n+1}+\cos\frac{4\pi}{2n+1}+\cdots+\cos\frac{2n\pi}{2n+1}$.
Show that $C_n^0-C_n^2+C_n^4-C_n^6+\cdots=2^{\frac{n}{2}}\cos\frac{n\pi}{4}$.
Show that
\begin{align*}
C_n^0-C_n^2+C_n^4-C_n^6+\cdots &=2^{\frac{n}{2}}\cos\frac{n\pi}{4}\\
C_n^1-C_n^3+C_n^5-C_n^7+\cdots &=2^{\frac{n}{2}}\sin\frac{n\pi}{4}
\end{align*}
Show that if $a, b, c$ are the lengths of the sides of a triangle, then the equation $$b^2x^2 +(b^2+c^2-a^2)x+c^2=0$$ does not have real roots.
Let $A, B,$ and $C$ be angles of a triangle. If $\cos 3A + \cos 3B + \cos 3C = 1$, determine the largest angle of the triangle.
Let $\alpha\in\Big(\frac{3\pi}{2}, 2\pi\Big)$. Simplify $$\sqrt{\frac{1}{2}-\frac{1}{2}\sqrt{\frac{1}{2}+\frac{1}{2}\cdot\cos 2\alpha}}$$
Given two segments $AB$ and $MN$, show that $$MN\perp AB \Leftrightarrow AM^2 - BM^2 = AN^2 - BN^2$$
In $\triangle{ABC}$, let $a$, $b$, and $c$ be the lengths of sides opposite to $\angle{A}$, $\angle{B}$ and $\angle{C}$, respectively. $D$ is a point on side $AB$ satisfying $BC=DC$. If $AD=d$, show that
$$c+d=2\cdot b\cdot\cos{A}\quad\text{and}\quad c\cdot d = b^2-a^2$$
Prove the triple angle formulas: $$\sin 3\theta = 3\sin\theta -4\sin^3\theta$$ and $$\cos 3\theta = 4\cos^3\theta - 3\cos\theta$$
Show that $\sin\alpha + \sin\beta + \sin\gamma - \sin(\alpha + \beta+\gamma) = 4\sin\frac{\alpha+\beta}{2}\sin\frac{\beta+\gamma}{2}\sin\frac{\gamma+\alpha}{2}$
$\displaystyle\frac{2\cos 2^n A+1}{2\cos A+1}=\prod_{r=1}^{n} (2\cos 2^{r-1}A-1)$
Given any $\triangle{ABC}$, show that $$\cos A + \cos B + \cos C = 1+4\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}$$
In $\triangle{ABC}$, show that $$\sin\frac{A}{2}=\sqrt{\frac{(p-b)(p-c)}{bc}}$$ where $p=\frac{a+b+c}{2}$ is the semi-perimeter.
Show that
$$\sec^2\alpha = 1 + \tan^2\alpha$$
$$\csc^2\alpha = 1 + \cot^2\alpha$$
Prove the identity: $\tan^2 x - \sin^2 x = \tan^2 x \sin^2 x$.
If $\cos x - \sin x = \sqrt{2}\sin x$, prove $\cos x +\sin x = \sqrt{2}\cos x$.
In $\triangle{ABC}$, show that
\begin{align*}
\sin 2A + \sin 2B + \sin 2C &= 4\sin A\sin B \sin C\\
\cos 2A + \cos 2B + \cos 2C &= -1-4\cos A\cos B\cos C
\end{align*}