Intereting Posts

Eigenvalues of a matrix with binomial entries
How many pairs of numbers are there so they are the inverse of each other and they have the same decimal part?
Is there a General Formula for the Transition Matrix from Products of Elementary Symmetric Polynomials to Monomial Symmetric Functions?
Convergence of power towers
A number system
Stress vector – Stress tensor
Does the ternary dot product have any geometric significance?
The probability that two vectors are linearly independent.
A finite set always has a maximum and a minimum.
Distinguishing between valid and fallacious arguments (propositional calculus)
Borel hierarchy doesn't “collapse” before $\omega_1$
Some question of sheaf generated by sections
How to start a math blog?
Given a prime $p\in\mathbb{N}$, is $A=\frac{x^{p^{2}}-1}{x^{p}-1}$ irreducible in $\mathbb{Q}$?
The limit of a sum

I’ve been trying to figure out how CEVA’s Theorem can be implemented in solving this problem, but I’m coming up short and cannot make any progress with this problem. The problem states;

A convex hexagon *ABCDEF* satisfies |*AB*| = |*BC*|; |*CD*| =

|*DE*|; |*EF*| = |*FA*|. Prove that the lines containing the altitudes of the

triangles *BCD*, *DEF*, *FAB* starting respectively at the vertices *C*, *E*, *A*

intersect at a common point.

Any advice or guidance is much appreciated!

- Intersection of two parabolae
- What are the postulates that can be used to derive geometry?
- To draw a straight line tangent to two given ellipses.
- Three Circles Meeting at One Point
- Prove converse Thales theorem, proportional sides imply parallel lines
- Seven points in the plane such that, among any three, two are a distance $1$ apart

- If ABCD is a square with A (0,0), C (2,2). If M is the mid point of AB and P is a variable point of CB, find the smallest value of DP+PM.
- What structures does “geometry” assume on the set under study?
- algorithm to calculate the control points of a cubic Bezier curve
- change of basis and inner product in non orthogonal basis
- Determine the centre of a circle
- Calculate the intersection points of two ellipses
- About the Riemann surface associated to an analytic germ
- $\angle ABD=38°, \angle DBC=46°, \angle BCA=22°, \angle ACD=48°,$ then find $\angle BDA$
- Maximum number of equilateral triangles in a circle
- Find the circle circumscribing a triangle related to a parabola

Draw the circles $k_B, \,\, k_D, \,\, k_F$ centered at the vertices $B, \,\, D,\,\, F$ and radii $BA=BC, \,\, DC=DE, \,\, FE=FA$ respectively. Then the altitude line $h_A$ of triangle $ABF$ throguh vertex $A$, the altitude line $h_C$ of triangle $BCD$ throguh vertex $C$ and the altitude line $h_E$ of triangle $DEF$ through vertex $E$ are the radical axes of the three pairs of circles $(k_F, k_B)$, $\,\, (k_B,k_D)$ and $(k_D, k_F)$ respectively. Therefore, by the radical axis theorem, the three radical axes $h_A, h_C$ and $h_F$ intersect at a common point.

Let P, Q, R be the feet of the said altitudes (in blue).

Then, red, purple, and green circles can be formed. Those blue lines happen to be the common chords. They concur at the common point X, the radical center.

If Ceva’s Theorem isn’t required, then I recommend Carnot’s Theorem (not to be confused with Carnot’s Theorem).

Let the pairs of edges meeting $B$, $D$, $F$ have respective lengths $b$, $d$, $f$, and let the altitudes from $A$, $C$, $E$ have lengths $a$, $c$, $e$. With $A^\prime$, $C^\prime$, $E^\prime$ the feet of those respective altitudes, we have …

$$\begin{align}

|\overline{BA^\prime}|^2 + |\overline{DC^\prime}|^2 + |\overline{FE^\prime}|^2 &= \left(\;b^2 – a^2\;\right) + \left(\;d^2 – c^2\;\right) + \left(\;f^2 – e^2 \;\right) \\[6pt]

&= \left(\;f^2 – a^2\;\right) + \left(\;b^2 – c^2\;\right) + \left(\;d^2 – e^2 \;\right) \\[6pt]

&= |\overline{FA^\prime}|^2 + |\overline{BC^\prime}|^2 + |\overline{DE^\prime}|^2

\end{align}$$

By Carnot, the perpendiculars to the sides of $\triangle BDF$ at points $A^\prime$, $C^\prime$, $E^\prime$ are concurrent. $\square$

- isolated non-normal surface singularity
- Minpoly and Charpoly of block diagonal matrix
- How to prove completeness of the Spherical Harmonics
- Analyzing whether there is always a prime between $n^2$ and $n^2+n$
- Exponential Generating Function of the numbers $r(n)$
- find the limit: $\lim_{n\to\infty}\int_{0}^{\infty} \frac{\sqrt x}{1+ x^{2n}} dx$
- Use the definition of “topologically conjugate” to prove that $F(x)=4x^3-3x$ is chaotic.
- Prove two pairs of subspaces are in the same orbit using dimension
- Is it wrong to use Binary Vector data in Cosine Similarity?
- Derivative of a function with respect to another function.
- Determining similarity between paths (sets of ordered coordinates).
- Associativity of norms in inseparable extensions
- Free idempotent semigroup with 3 generators
- Let $p$, $q$ be prime numbers such that $n^{3pq} – n$ is a multiple of $3pq$ for all positive integers $n$. Find the least possible value of $p + q$.
- Is Thomae's function Riemann integrable?