For the line given by, y = 3x , find the slope of a line that is: a) Parallel to the given line: m_{parallel} = b) Perpendicular to the given line: m_{perpendicular} =

A circle is circumscribed by a parallelogram. prove by using tangent property that the parallelogram is a rhombus. I tried to prove that the adjacent sides of the parallelogram are equal but I lack idea to apply tangent property

Prove by vector method that the diagonals of a rhombus bisect each other. Also, show that they bisect each other at right angles

My Attempt:

Let us consider a rhombus $OACB$ where $\overrightarrow{OA}=\overrightarrow{a}$ and $\overrightarrow{OB}=\overrightarrow{b}$. Then,
$\overrightarrow{OC}=\overrightarrow{OA}+\overrightarrow{AC}$
$\overrightarrow{OC}=\overrightarrow{OA}+\overrightarrow{OB}$
$\overrightarrow{OC}=\overrightarrow{a}+\overrightarrow{b}$
Similarly for $\overrightarrow{AB}$ we can write $\overrightarrow{AB}=\overrightarrow{b}-\overrightarrow{a}$.

Consider the following wave equation:
$\begin{array}{rl}{u}_{tt}-u_{xx}& =0,0<t<tx,k>1\\ u{|}_{t=0}& ={\varphi }_0(x),x\ge 0\\ u_t{|}_{t=0}& ={\varphi }_1(x),x\ge 0\\ u{|}_{t=kx}& =\psi (x)\end{array}$
In which ${\varphi }_0(0)=\psi (0)$.

The problem is that on part of the sector where $t>x$ d'Alembert's formula isn't applicable. It seems we will have to do some sort of extension or reflection, but how to start?

I hear about axioms in set theory and postulates in geometry, but they seem like the same thing. Do they mean the same thing but then are used in different instances or what? Is one word more applicable in one case more than the other? I never hear of axioms in geometry or postulates in set theory. Are axioms more formal and postulates used more informally?

Triangle $ABC$; $AB=c,BC=a,AC=b$; angle bisector of angle ($c$,$a$) cuts $A$$C$ in point $E$
Why is the following true?
$|AE|=\frac{bc}{a+c}$
Where does that come from?

I'm trying to prove some properties of sequence spaces. I already know that the space $l^{\mathrm{\infty }}$ of all bounded sequences isn't an inner product space, isn't separable but it is complete with sup norm.
The space c of all convergent sequences is also complete, as a closed subspace of $l^{\mathrm{\infty }}$, and so is $c_0$ - the space off all sequences with entries equal to zero from some point on.
I also know that both $c$ and $c_0$ are separable.
My question is - are $c$ and $c_0$ inner product spaces. Should I look for an example of sequences which don't satisfy the parallelogram law.

Prove: If the sum of the angles of a triangle is a constant n, then $n=180$ and thus the geometry is Euclidean