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### Find a synthetic solution: $T(12^{\circ}, 24^{\circ}, 54^{\circ}, 18^{\circ}, 30^{\circ})$

Hint: Triangle ABC is isosceles and $\angle BAC=72^o$. As can be seen in figure the bisector of angle ACB meets side AB at E . We have $\angle ECD=\angle ACD=18^o$. Triangle ACE is isosceles(why ?), ...
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### Number of chords to be drawn to return the starting point

Method using modular arithmetic Notice that $\triangle AOB$ is isosceles and therefore $\angle AOB = 105^\circ$. Each successive chord adds $105^\circ$ to this angle so we want to know when this would ...

### Find the volume of the pyramid below

You can find the result by subtracting from the volume of the cube the volumes of other five pyramids, whose volume is easy to find because their bases lie on the faces of the cube: pyramid $HIJG$ (...
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### What does 3-regularity mean in the context of this diagram of a fullerene?

A graph is called 3-regular if every vertex (carbon atom) has 3 edges (bonds). More generally, in an n-regular graph every vertex has degree n. Another name for a 3-regular graph is a cubic graph. So ...
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### Is there any way to add spherical vectors without converting?

You can do it using formulas that are used in great-circle navigation. First, find the angle $\theta$ between the two vectors, which is equivalent to finding the distance between two points on a ...
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Accepted

### Find the measure of the smallest angle that determine the diagonals of a quadrilateral

Comment: as you see in figure if triangle is equilateral then minimum of $\theta$ is ($\theta=24^o$)(triangle DEF and quadrilateral FIJE). ($\theta=6^o$) is possible if triangle is isosceles(...
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First of all, note that $$(x_{i+1} - x_i)(y_{i+1} + y_i) = x_{i+1} y_{i+1} + x_{i+1} y_i - x_i y_{i+1} - x_i y_i.$$ When you add this up over all the pairs of vertices $(x_i, y_i)$ and $(x_{i+1}, y_{... • 86.6k -1 votes Accepted ### Ratio between the area of a triangle and a trapezoid This is the answer to your problem. Correct answer is$(C)=\dfrac{1}{7}$• 1,756 0 votes ### Ratio between the area of a triangle and a trapezoid Hint:$A_{ABCD}=2A_{AEB}+A_{BEC}$,$A_{BEC}=\frac{1}{3}A_{AEB}$,$A_{AEC}=\frac{1}{3}A_{AEB}$. • 8,417 3 votes ### Relating energy integral to the existence of a Frostman measure If$B(x,r)$is disjoint from$A$,$\,$then$\nu(B(x,r)=0$. Otherwise, there is some point$y \in B(x,r) \cap A \,. \,$Then$B(x,r) \subset B(y,2r), so $$(2r)^{-s} \nu(B(x,r)) \le \int_{B(y,2r)} ... • 16k 0 votes ### Constant AM,GM,HM circles for segments through a point on negative power Circles Without loss of generality, by translational, rotational and scaling symmetries, we can assume that O=(0,0), the power line makes an angle \alpha with respect to the x-axis and the circle has ... 1 vote ### The points (0, k),(0,-k),(k, 0), and (-k, 0), where k \neq 0, comprise the vertices of a square. Why is the side-length not 2k? First, what is the error you're making? Well, I tend to doubt that you've actually drawn a picture of the square you have in mind, with the vertices carefully labelled. From what you're saying it ... • 85.6k 1 vote ### The points (0, k),(0,-k),(k, 0), and (-k, 0), where k \neq 0, comprise the vertices of a square. Why is the side-length not 2k? Trivial answer. We have a square \rightarrow it has 4 sides having equal (Euclidean) length. Then, by Pythagorean Theorem, the length of one side is given by \sqrt{(k-0)^2+(k-0)^2}=\sqrt{2(k-0)^2}... • 159 0 votes ### Finding the length of one side of cyclic trapezium if given length of two adjacent side and the diagonal between them. Draw a point E on DC such that BE is perpendicular to DC. Let EC = x. Since this is a cyclic trapezium, its also isosceles, and hence BC=5 and DE=4+x. Hence,$$\begin{aligned}BE^2=BD^2-... 1 vote Accepted ### Find an angle in a picture of trapezoids Sorry that I do not yet know how to draw figure using Latex or Mathjax. I can only describe how I approach the problem. LetAB$be the slant edge that make angle$\theta$with the ground. In the ... 1 vote ### Find the exact value of$\frac{BD}{AB}$From triangle$ADC$we get$AD = 2\cdot AC\cdot \sin(24)$. Now use the law of cosine in triangle$ABD$to achieve (after some manipulations) $$BD^2 =2 AC^2.$$ • 16.8k 1 vote ### prove that this expression equals the area of a specific triangle Let$a = 2, A = \frac {\pi} 4, B = \thetaThen, $$\frac{b}{\sin \theta} = \frac{2}{\frac{\sqrt{2}}2} \implies b = 2\sqrt 2 \sin \theta$$ Then, $$\sin C = \sin(A+B) = \frac {\sqrt{2}}2 (\sin \theta ... • 224 0 votes ### Studying Euclidean geometry using hyperbolic criteria Charles' Segal's answer gives an answer about how you could do it if you lived in 3 dimensional hyperbolic space. Here's an answer on how you could do it if you lived in a 2-dimensional hyperbolic ... • 780 0 votes ### How can I create a function which takes input and gives new coordinate X, Y which has rotation in degree of some angle? If a point, (X, Y), has distance r from the origin and lie on a line at angle \theta from the line y= 0, then X= r cos(\theta) and Y= r sin(\theta). After a rotation through 60 degrees= pi/3 ... • 734 1 vote ### geometrical/physical interpretation of multiplication of real numbers (including negative) It is important to understand how the "Ancient civilizations" were defining the product of two positive quantities a and b. @Lee Mosher has given such a definition. Another one is ... • 66.5k 0 votes ### What's the 2d-object that's in 1:1 correspondence with all lines in \Bbb R^2? Wanted to present a "twistorial" perspective. The oriented affine lines of \mathbb{R}^2 may be identified with the unit circle's tangent bundle TS^1: given any tangent vector t to a ... • 16k 1 vote Accepted ### Proving that when the normal unit vector N is parallel to the position vector \gamma(t) then the image \gamma(I) is part of circle Vector N(t) is perpendicular to \gamma'(t), hence:$$ \gamma'(t)\cdot\gamma(t)=0\ \implies {d\over dt}\big(\gamma(t)\cdot\gamma(t)\big)=0 \ \implies \|\gamma(t)\|=\text{constant}. $$• 42.8k 1 vote Accepted ### Is this a valid proof for the pythagorean theorem? Your proof is correct and is the standard one, but you're being overly clear about what you're doing (you don't need to do everything in really small steps). Indeed it does suffice to just say that if ... • 3,586 6 votes Accepted ### What's the 2d-object that's in 1:1 correspondence with all lines in \Bbb R^2? The brief answer to your question is: an open Möbius band. In more detail, let's embed \mathbb R^2 into \mathbb R^3 by the map (x,y) \mapsto (x,y,1). Using this embedding, I am going to abuse ... • 102k 0 votes ### Equation of a cylinder with a profile / ellipsoid Btw, if the equation is sought for finding volume of a Barrel it can be done by integration with attention to limits, resulting in the formula$$ V= \frac{\pi h }{12}\cdot (2D^2+d^2)$$where D is ... • 36.9k 1 vote ### Equation of a cylinder with a profile / ellipsoid Consider the blue curve first In my notation the small radius is r=0.01 and the large radius is R=0.09 It is an arc, of radius R and center at x=R - r I use the parameter angle \theta for ... • 10.9k 1 vote ### Equation of a cylinder with a profile / ellipsoid That would probably be called a "barrel shape". The equation depends on the type of curve you choose for the barrel. If it's a circular curve, if the y axis is along the axis of symmetry ... • 8,200 0 votes Accepted ### The lemniscate of Euler's \textit{elastica}: precise determination of a characteristic constant In the comments, @TravisWillse provides a very useful link to Djondjorov et al.'s "Explicit parameterization of Euler’s elastica." The authors express the limits of the lemniscate as \left(\... 1 vote Accepted ### Find the dimension of the interior bin of a rectangular storage container We know from the volume that lwh=240, and that 2(l+w)=40 from the perimeter. We can rearrange the latter as l=20-w. Now, we substitute this expression for l into the equation for volume. We ... 3 votes ### Trapezoid height from one base to other base given tangent circle I am using h as defined by the OP. The altitude of M to DC is also h and has base point I on DC. As F is the midpoint to DM using scaling the altitude of F to DC is h/2. Thus EF=... • 1,318 4 votes Accepted ### Trapezoid height from one base to other base given tangent circle It's easy to show that the horizontal component of M is 6, and the vertical component is h/2 (h is the height AH). Then$$DM=\sqrt{6^2+\frac{h^2}4}$$Also easy to see that the vertical ... • 33.4k 0 votes ### Area of Intersection: Rectangle and Two Circles If the offset is (A), then the area of the shaded region is also (A). Analysis follows (no Calculus required). The picture below provides three figures. In Figure 1, for clarity, I have ... • 24.7k 2 votes ### If \vec a,\vec b,\vec c be three vectors such that |\vec a|=1,|\vec b|=2,|\vec c|=4 and then find the value of |2\vec a+3\vec b+4\vec c| You have$$2a + 3b +4c = 3(a+b+c) -a+c.$$therefore$$\begin{aligned} \lVert 2a + 3b +4c \rVert^2 &= 9 \lVert a + b +c \rVert^2 + \lVert a \rVert^2 + \lVert c \rVert^2 - 6 \lVert a \rVert^2 - 6 a \... 3 votes ### If\vec a,\vec b,\vec c$be three vectors such that$|\vec a|=1,|\vec b|=2,|\vec c|=4$and then find the value of$|2\vec a+3\vec b+4\vec c|$Recall that$\|\vec x\|^2 = \vec x \cdot \vec x\$. Then \begin{align*} \|2\vec a + 3\vec b + 4\vec c\|^2 &= (2\vec a + 3\vec b + 4\vec c) \cdot (2\vec a + 3\vec b + 4\vec c) \\ &= 4\|\vec a\|^... • 11k 2 votes ### geometrical/physical interpretation of multiplication of real numbers (including negative) I believe the geometrical interpretation still work, if you take the signed area of the rectangle. So, let's define an "oriented" rectangle as a rectangle with a particular orientation of ... • 16.7k 3 votes Accepted ### geometrical/physical interpretation of multiplication of real numbers (including negative) There are a couple of different ways to express multiplication using geometry. One somewhat physical interpretation of multiplication involves stretching/compressing/reflecting the number line. For ... • 102k 1 vote ### Intersections of lines formed by the pentagon are collinear. This is not yet the kind of proof based on Pappus and Desargues that you're after. But it might shed some light so I'm posting it now and may add more later in an edit or a separate answer. This is a ... • 38.6k 1 vote Accepted ### Can we calculate the sides of a right triangle knowing only its three angles and the sides of a similar triangle? No. You can apply an arbitrary scaling transformation to a triangle. Doing so will preserve similarity and all angles, but change lengths. Which means that you can't determine lengths if all you have ... • 38.6k 0 votes ### Rectangle where E is a point on side and finding length of angle bisector of \angle BED Let drop altitude AF to BE. Then triangles ADE and AFE are congruent right triangles. Then AF is 4. Using Pythagoras theorem one can find BF=\sqrt{AB^2-AF^2}=3. Angles BAE and AEB are ... • 3,663 1 vote ### Rectangle where E is a point on side and finding length of angle bisector of \angle BED Alternative (less elegant) approach: \underline{\text{Tools:}} \displaystyle \tan (r + s) = \frac{\tan(r) + \tan(s)}{1 - [\tan(r)\tan(s)]}. \displaystyle \tan (r - s) = \frac{\tan(r) - \tan(s)}{... • 24.7k 1 vote ### Equation of a line passing through the intersection of 2 lines This is dual to the situation where two points \,p_0\, and \,p_1\, in any affine (or vector) space. determine a unique line. In this case, the set of points of the line is parametrized by the ... • 30.1k 1 vote Accepted ### Rectangle where E is a point on side and finding length of angle bisector of \angle BED From your angles labeled \beta and \gamma it follows BAE is an isosceles triangle. Therefore BE=5. Plotting this on the coordinate plane with B=(5,4) and E=(x,0), then we have\sqrt{(5-x)...
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