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Case 1: Well done; you could have just specified that $VC=VA(=2x\sqrt3)$ Case 2: Well done; you could have quoted the angle bissector theorem and perhaps write more classicaly $\frac{V\color{red}A}{V\... • 1,971 2 votes Accepted ### The triangle$ABC$is right-angled in$A$. Prove that the inequality$(1-\sin B)(1-\sin C)\leq \frac{{(\sqrt{2}-1)}^2}{2}$holds. WLOG the hypotenuse is 1. Let the legs be$a,b$. Denote$ab = x$. $$(1-a)(1-b) = \frac{a^2b^2}{1+a+b+ab} \le \frac{a^2b^2}{1+2\sqrt {ab}+ab} = \left(\frac{x}{1+\sqrt x}\right)^2$$ Now, consider: $$\... • 3,601 2 votes ### The triangle ABC is right-angled in A. Prove that the inequality (1-\sin B)(1-\sin C)\leq \frac{{(\sqrt{2}-1)}^2}{2} holds. As it was pointed out in other answers and comments, this amounts to determining the maximum value of f(x)=(1-\sin x)(1-\cos x) for x \in [0, \frac{\pi}{2}]. Since f is differentiable, the ... • 20.9k 1 vote ### The triangle ABC is right-angled in A. Prove that the inequality (1-\sin B)(1-\sin C)\leq \frac{{(\sqrt{2}-1)}^2}{2} holds. \sin B = \dfrac{b}{a} , \sin C = \dfrac{c}{a} , so (1 - \sin B) (1 - \sin C) = \dfrac{(a - b)(a - c)}{a^2} We can take a = 1 without loss of generality because we talking about angles here not ... • 21k 2 votes ### The triangle ABC is right-angled in A. Prove that the inequality (1-\sin B)(1-\sin C)\leq \frac{{(\sqrt{2}-1)}^2}{2} holds. Hint : It is a right angle triangle so, B + C = 90^{\circ}. So, this can be re-written as$$(1-\sin B)(1-\cos B)$$I hope you will be proceed from to get maxima at B=45^{\circ} and then the final ... 0 votes ### Determine position of sound source from arrival times of a blip I think this is similar to your idea but using the concept of circles. Since we do know the distance of the source from each receiver, we can write the eqn of 3 circles such that each receiver is the ... 0 votes ### The equation of a line reflected about another line Let l be the line ax+by+c=0. Let m be the line dx+ey+f=0. Let r be the reflection of m in l. When we reflect an equation in l, to get the new equation, simply replace each x with x-... • 220 5 votes ### How to calculate the area of a projected path onto a plane? (Bee traveling problem) I think this problem is intended to be a purely logical problem with very little vector geometry. For starters, the problem states (without proof) that there IS a plane that passes through the ... • 3,193 3 votes ### How to calculate the area of a projected path onto a plane? (Bee traveling problem) Adopt a Cartesian coordinate system such that the bee begins at the origin and east/north/up is in the direction of increasing x/y/z, respectively. Then the trajectory of the bee is a series of ... 0 votes ### Finding the coordinates of a moving point along the surface of a sphere. The standard directions at any point P_1( R , \theta_0, \phi_0) on the surface of the sphere are u_1 = \begin{pmatrix} - \sin \phi_0 \\ \cos \phi_0 \\ 0 \end{pmatrix} u_2 = \begin{pmatrix} - \... • 21k 1 vote ### How to calculate the area of a projected path onto a plane? (Bee traveling problem) Following is the code I developed to solve this problem. The square of the area output by the program is A^2 = 12 ... • 21k 2 votes ### Volume of a great icosahedron First of all, @ChrisLewis is totally right. The area of a great icosahedron can be described as the area of a small stellated dodecahedron with 5\cdot 12=60 congruent irregular tetrahedra carved out ... • 1,592 2 votes ### geometry problem where the solution involves the use of some properties of complex numbers in geometry Let me first replace this issue in a classical context. Let t:=a+b+c \tag{1} Alignment relationships 2O+H_k=3G_k give :$$\begin{cases}h_1&=&a+b+m\\ h_2&=&b+c+m\\ h_3&=&c+a+... • 81.2k 5 votes ### Volume of a great icosahedron Let us rescale the great icosahedron so that its true vertices have as coordinates cyclic permutations of$(\pm1,0,\pm\varphi)$where$\varphi$is the golden ratio$\frac{\sqrt5+1}2$, so that its true ... • 102k 0 votes ### Volume of a great icosahedron I'll find the point$B$from the following illustration using the regular icosahedron from this answer. (sorry to every reader for the godawful "graphic design"...) Namely, it can be taken ... • 1,232 17 votes Accepted ### What's wrong with this derivation of the volume of a hemisphere? The thing to keep in mind with disk integration is the fact that the variable of integration represents the thickness of each disk. You are trying to integrate the volume of a half-sphere, using: $$\... • 1,109 0 votes ### Investigating the shortest line segment connecting two disjoint ellipses Just solve the following convex optimization problem: \min \, \| r_1 - r_2 \|^2 subject to: (r_1 - C_1)^T Q_1 (r_1 - C_1) \le 1 (r_2 - C_2)^T Q_2 (r_2 - C_2) \le 1, which is classified as a ... • 2,313 0 votes ### Intersection of two ellipses at exactly 2 points We can solve explicitly this problem with the use of tangency. Given two ellipses$$ \cases{ a_1 x^2+b_1 x y + c_1 y^2 - k = 0\\ a_2 x^2+b_2 x y + c_2 y^2 - l = 0 } $$eliminating y we have$$ p(x) =... • 32.5k 13 votes ### What's wrong with this derivation of the volume of a hemisphere? Your problem seems to be that you don't have a clear concept of how to integrate the volume of an object by stacking disks on each other. In order to compute a volume by stacking disks, you need the ... • 97.7k 1 vote ### Does this spiral, formed from similar right triangles arranged around a point, have a name? This is not just a class of logarithmic spiral, it is exactly a logarithmic spiral. If we call the angle of the triangle$\Delta\theta$, and growth rate, call it$q=1/\cos\Delta\theta$, the flair ... • 7,224 2 votes Accepted ### How can I calculate position of a point after it moved towards another point? Here is an image explaining everything. It's just the Pythagorean Theorem. • 124k 2 votes Accepted ### What is the conditiones to have the plane$(ABC) $and$(DEF) $perpendicular? I'll assume that$A$,$B$and$C$are pairwise different and that the same holds for$D$,$E$and$F$(otherwise the question is meaningless). We have that$$(ABC)\perp (DEF)\iff \textbf{u}\perp \... 28 votes ### What's wrong with this derivation of the volume of a hemisphere? You're computing the volume of a cone, not a hemisphere! More precisely, your integral gives the volume obtained by rotating the triangle$0 \le y \le x \le r$about the$x\$-axis. The hemisphere, on ...
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