Recent content by anuttarasammyak

I find the above is for polygon which includes centern of circumcircle in it. As shown in #61 including polygon which does not include the center, the condition is $$(\sum_{i=1}^m \sin^{-1}\frac{f(L_i)L_i}{d})(\sum_{i=1}^m \sin^{-1}\frac{L_i}{d}-\pi)=0$$ defined for d > the longest ##L_i## and...

There are m rods length of each does not surpass sum of the others. With these rods we can make m polygons which has a circumcircle. It seems to hold.

Where d is diameter of the circle re:#31. Correction: Any side must be shorter than sum of all the other sides. Heron and Brahmagupta area must be real number.

In generalc case of m polygon $$\sum_{i=1}^m \sin^{-1} \frac{L_i}{d}=\pi$$ seems to hold with choice of parameter d for any ##{L_1,L_2,..,L_m}##. So there seems to exist a circumcircle.

$$2A_{abcd}=(ab+cd)\sin\theta = (ab+cd)\sqrt{1-(\frac{a^2+b^2-c^2-d^2}{2(ab+cd)})^2}$$ $$=\frac{1}{2}\sqrt{4(ab+cd)^2-(a^2+b^2-c^2-d^2)^2}$$ $$4A_{abcd}=\sqrt{(b+c+d-a)(c+d+a-b)(d+a+b-c)(a+b+c-d)}$$

Thanks, I got it. I noticed that we can get unike circumcircle radius for any given a,b,c,d >0 by transforming angles of quadrilateral. $$e=\sqrt{\frac{ab(c^2+d^2)+cd(a^2+b^2)}{ab+cd}}$$ Circumradius of triangle sides of which are a,b,e is...

Thanks for the teachings. I might have failed for null vevtors in #39. Putting d＝0 in the general solution I got familiar formula for triangle. It is a good check for me. I would likt to confirm that the general solution does not depends on whether quadrilateral contains O which is center...

continued form #31 We get three equations $$\begin{pmatrix} L_2 & L_1 & -L_4 & -L_3 \\ L_3 & -L_4 & L_1 & -L_2 \\ L_4 & -L_3 & -L_2 & L_1 \\ \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3\\ x_4\\ \end{pmatrix} = \begin{pmatrix} 0 \\ 0\\ 0\\ \end{pmatrix}$$ where...

Instead let us use cos (a+b)=cos a cos b - sin s sin b for the case that quadrilateral contains conter of the circle $$x_1x_2+x_3x_4=L_1L_2+L_3L_4:=A$$ $$x_2x_3+x_4x_1=L_2L_3+L_4L_1:=B$$ where $$x_i:=\sqrt{d^2-L_i^2}$$ In the process of deletion of a x, we observe that x's are linear dependent...

$$\sum_{n=1}^4 \sin^{-1} \frac{L_n}{d}=\pi$$ where d is diameter of circle and l_i are length of sides of quatrilateral. Using sin(a+b)=sin a cos b + cos a sin b, and scaling ##a_n=\frac{L_n}{d}## We obtain three equations, as far as the quadrilateral contains center of the circle...

You have four slices of one kind. Cut one slice to equal three pieces. Now you have 12 pieces > 9 people. Another approach 3 people have 2 cakes. 6 people have 3 cakes. Let the three, say a b c, choose their favorite by two turns, ordered abc cba. Then let the six, say d e f g h i, choose...

Forward. I am poor at finding the puzzle here.

I will try. For 1kg liquid water at boiling temperature T under constant pressure p to become steam gas completely, we provide heat which is divided into latent heat and work the system does. $$Q=L+W=L+p(V_g-V_l)$$ In approximation that ##V_l << V_g## for occupied volumes of liquid water and...

As far as I see in the web and AI answers they use Clapeyron Clasusius equation to explain it. I wonder if it is what expected by the homework teacher.

It is a Fourier series which has abundant applications. it has relation with delta function. [EDIT] Though a bit different from your formula, complex Fourier series for Heaviside step function, H(x)=0 for x<0 H(0)=1/2 H(x)=1 for x>0 in period 2##\pi##, is...