# Questions tagged [geometry]

For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, and angles.

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### Finding area of a triangle from incircle of a square and given intersection lines.

The incircle of square $ABCD$ touches sides $\overline{AB},$ $\overline{BC},$ $\overline{CD},$ and $\overline{DA}$ at $P,$ $Q,$ $R,$ and $S,$ respectively. Point $E$ lies on arc $PQ$ of the incircle. ...
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### Find radius of largest circle within ellipse $\frac{x^2}{9} + \frac{y^2}4 = 1$ with their intersection only at $(3,0)$

An ellipse is defined by the equation $$\frac{x^2}{9} + \frac{y^2}4 = 1$$ Compute the radius of the largest circle that is internally tangent to the ellipse at $(3,0),$ and intersects the ellipse ...
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### Maximum social distancing puzzle

A spherical planet of radius $R$ is inhabited by $n\ge 2$ aliens. Due to the emergence of a novel deadly virus, the aliens must practice social distancing, staying away as far as possible of each ...
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### Parametric curve of a spiral with straight and semicircular parts

I need to design a fully parameterized spiral like this where I can choose/change the central part length and the radius of the innermost curve. I understand that it will be somehow related to the ...
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### Test if a point on a hexagonal lattice falls on a specified superlattice?

Based on previous answers (1, 2, 3) integers $i, j$ produce a hexagonal lattice using $$x = i + j/2$$ $$y = j \sqrt{3} / 2.$$ From a point $k, l$ I can make a superlattice from integers $I, J$ ...
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### Look at equation $6xy - 30x + 20y - 100 = 0$. Do transformation coordinate axes to change the cone section to standard form!

The full question: Look at equation $6xy - 30x + 20y - 100 = 0$. Do transformation coordinate axes to change the cone section to standard form? Sketch out the cone sections graph Its my task from my ...
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### Conjugate diameters of ellipsoid

I am familiar with ellipses,but am seeking to find the condition for 2 diameters of an ellipsoid to be conjugate.Any help or references would be appreciated.
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### Obtaining condition for existence of two Chords through a single point with their Mid-points on the $x$-Axis

Given a circle $x^2+y^2=\alpha x + \beta y$, obtain the condition on $\alpha$ and $\beta$, if two distinct chords can be drawn through the point $(\alpha, \beta)$ such that their mid-points lie on the ...
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### Is there a geometric proof for distributivity of integer addition/multiplication and other similar properties?

I am aware that commutativity, associativity and distributivity of integer addition and multiplication follow from their standard set theoretic definitions but I am looking for something suitable for ...
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### In the case of symmetric matrix diagonalization, why we take orthonormal eigenvector matrix?

For a matrix $A$ we can use diagonalization using this formula: $$A = X D X^{-1}$$ where $X$ is a matrix containing eigenvectors in the columns and $D$ is the diagonal matrix containing eigenvalues. ...
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### Finding the area of a fraction of a circle defined by points on a grid.

A grid is shown below, where the shortest distance between any two points is 1. Find the area of the circle that passes through points $X,$ $Y,$ and $Z.$ Can it be assumed that the arc created over ...
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### Hint as to how to formalize that the curvature of a curve inside the unit circle is bounded below by the curvature of the circle

I am trying to self teach differential geometry and to that effect I am trying to do the Homework in the MIT open course. The specific question I am struggling with is: Let $c$ be a regular curve ...
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### Are the two triangles similar?

A book I was answering was asking for the center of enlargement and the scale factor of the two similar figures.But I don't think the two triangles are similar. They corresponding sides don't have the ...
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### Convert linear distance to steering angle

I need to calculate the angle of the front steering wheel using a collapsible piston(linear sensor). 'x' is used to represent the length in inches of the movable part of the sensor and is the ...
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### Chord of contact to a second degree homogeneous curve [closed]

How can I prove that equation of chord of contact to a second degree homogeneous curve of the form of ax² + by² + 2hxy + 2gx + 2fy + c is T = 0 also that equation of a chord with a given middle point ...
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### Right circular cone shortest distance question from a junior high school st. [duplicate]

Ant's shortest distance that it can travel Can you show me the answer in the cone unfolded. I would be glad if you guys can show me the solution in a simple way because I am a high school student.
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### What logic can I use to determine the complete overlap of multiple polygons?

I need to measure the 2D space that multiple arbitrary polygons occupy. To do so, I need to add the area of all polygons and subtract the overlap. I already have written a function that takes in only ...
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### Are there geometries in which points have finitely many parallel lines?

In Euclidian geometry the Parallel Postulate holds that given a parallel line as a reference there is one parallel line through any given point. In elliptic geometry there are no parallel lines. In ...
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### Distance between two generated lines

Write the equations of the rectilinear generatrices of the hyperbolic paraboloid $\frac{x^2}{p}-\frac{y^2}{q}=2z\ \ ,p,q>0$. Out of these rectilinear generatrices select those which are parallel to ...
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### Intersection of oblate spheroid $\frac{x^2+y^2}{R_e^2}+\frac{z^2}{R_p^2} =1$ and plane $n_xx+n_yy+n_zz=0$

To calculate the distance between two points on Earth, I used 3 different approaches. For small distances, I used the Euclidean distance. For medium distances, I used the arc length on the circle ...
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### When does a convex set have a unique outward normal direction?

Let $C$ be a closed, nonempty, and convex set (in a real Hilbert space $\mathcal{X}$), and let $c\in C$ be a point on its boundary. When will the normal cone $N_Cc$ have a unique (nonzero) direction? ...
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### Finding fraction of a volume by visualization of cube being cut into separate pieces

Cube $ABCDEFGH$ is cut into four pieces by cutting along planes $BCHE$ and $BDHF.$ Find the fraction of the volume occupied by the piece containing the vertex $A.$ I'm not really able to visualize ...
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### Finding triangle side lengths given altitudes or lengths of sectioned triangles within triangle.

In right triangle $ABC,$ $\angle C = 90^\circ.$ Let $P$ and $Q$ be points on $\overline{AC}$ so that $AP = PQ = QC.$ If $QB = 67$ and $PB = 76,$ find $AB.$ How do I use ratios and given side lengths ...
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### sector area of a circle [closed]

A little circle and big circle share the same center. The radius of the little circle is 4 cm and the radius of the big circle is 9 cm. What is the area of the shaded region shown if the measure of ...
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### Find the shaded area, given the sides:

The given sides were $AC=BD=25$, $AD=BC=15$, and $DC=7$. No other explanations were given to this problem. I tried to connect A and B to form an isosceles trapezoid then tried to work around with ...
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### Can a polygon be monotone with respect to precisely one direction?

This question is first exercise in section 2.2.3 in book "Computational Geometry in C". I want to check if such a polygon exists, and if it does, then prove that it is indeed monotone with respect to ...
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### Question from PRMO 2019.

An ant leaves the anthill for its morning exercise. It walks 4 feet east and then makes a 160° turn to the right and walks 4 more feet. It then makes another 160° turn to the right and walks 4 more ...
I am solving this question: Let $ABC$ be an acute angled triangle and $CD$ be the altitude through $C$. If $AB=8$ and $CD=6$, find the distance between the midpoints of BC and AD. So I observed that ...