Intereting Posts

Proof $(1+1/n)^n$ is an increasing sequence
When can't a real definite integral be evaluated using contour integration?
Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $a^2 + b^2 + c^2 \ge a + b + c$.
Number of ways of spelling Abracadabra in this grid
consequence of mean value theorem?
Product of Riemannian manifolds?
Fermat's 2 Square-Like Results from Minkowski Lattice Proofs
Show: $f'=0\Rightarrow f=\mbox{const}$
Solving an equation with LambertW function
How do you calculate the decimal expansion of an irrational number?
Show that if $fd'=f'd $ and the pairs $f, d $ and $f',d' $ are coprime, then $f=f' $ and $d=d' $.
2D point projection on an ellipse
Galois theory: splitting field of cubic as a vector space
Reducibility over a certain field.
How to calculate the improper integral $\int_{0}^{\infty} \log\biggl(x+\frac{1}{x}\biggr) \cdot \frac{1}{1+x^{2}} \ dx$

Can anyone prove the Ptolemy inequality, which states that for any convex quadrulateral $ABCD$, the following holds:$$\overline{AB}\cdot \overline{CD}+\overline{BC}\cdot \overline{DA} \ge \overline{AC}\cdot \overline{BD}$$

I know this is a generalization of Ptolemy’s theorem, whose proof I know. But I have no idea on this one, can anyone help?

- geometry question on incenters
- Proof of Pythagorean theorem without using geometry for a high school student?
- How do you determine if a point sits inside a polygon?
- Integral points on a circle
- How to calculate volume given by inequalities?
- Intuitive geometric explanation: existence of eigenvalue in odd dimension real vector space.

- A geometric reason why the square of the focal length of a hyperbola is equal to the sum of the squares of the axes?
- Prove the inequality $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3}$
- Calculate polyhedra vertices based on faces
- Prove $F(n) < 2^n$
- What is the difference between a variety and a manifold?
- Reflected rays /lines bouncing in a circle?
- Drawing a thickened Möbius strip in Mathematica
- Find the minimum value of $P=\sum _{cyc}\frac{\left(x+1\right)^2\left(y+1\right)^2}{z^2+1}$
- Is it possible to draw this picture without lifting the pen?
- The area of the region $|x-ay| \le c$ for $0 \le x \le 1$ and $0 \le y \le 1$

Represent the four vertices as the complex numbers $a, b, c, d$. Notice that

$$ (a-b)(c-d) + (a-d)(b-c) = (a-c)(b-d),$$

which you can simply multiply out. Then we must have that

$$\begin{align}

|(a-b)(c-d) + (a-d)(b-c)| &= |(a-c)(b-d)| = |a-c||b-d| \\

|(a-b)(c-d)| + |(a-d)(b-c)| &\geq \\

|a-b||c-d| + |a-d||b-c| & \geq

\end{align}$$

which is equivalent to your question. To pass from the first line to the second, we used the triangle inequality. To pass from the second to the third (and to get the second equality on the first line), we used that absolute values are multiplicative.

In general, many elementary geometry facts are bared easily with complex numbers, or vectors in general for higher dimensions (but being able to just multiply is pretty nice).

- What is the Fourier Transform of $f'(x)/x$
- A improper integral with complex parameter
- An interesting identity involving powers of $\pi$ and values of $\eta(s)$
- Proj construction and ample dualizing sheaf
- Another proof of uniqueness of identity element of addition of vector space
- Profinite completion is complete.
- Relation between Heaviside step function to Dirac Delta function
- When $\min \max = \max \min$?
- What are the differences between Hilbert's axioms and Euclid's axioms?
- Do the digits of $\pi$ contain every possible finite-length digit sequence?
- Can anyone help me with a solution?
- Produce a sequence $(g_n):g_n(x)\ge 0$ and $\lim g_n(x)\neq 0$ but $\int_{0}^{1} g_n\to 0$
- Can infinitely many primes lie over a prime?
- Is $Z(R)$ a maximal ideal?
- Unilateral Laplace Transform vs Bilateral Fourier Transform