# Questions tagged [geometry]

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

31,119 questions
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### Line through a a given point in the first quadrant of the coordinate plane to form a triangle [closed]

Consider a straight line with negative gradient passing through the positive quadrant (where all co-ordinates are positive, or the first quadrant) of the co-ordinate plane and intercepting the $x$ and ...
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### distribution of bisecting great circles

For any point on the globe, I believe there is (by the mean value theorem) at least one great circle containing that point and dividing the world's land area (or water mass, or population, whatever) ...
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### a problem about the incircles of two triangles that the orthocenter formed.

See below diagram. $H$ is the orthocenter of an acute triangle $ABC$ where $AB \neq AC$. The circle centered at $I$ and the circle centered at $J$ are the incircles of triangles $ABH$ and $ACH$. $XY$ ...
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### How many uniform polytopes are there in higher dimensions?

I am not really interested in the exact numbers, but more in the richness of the class of uniform (convex) polytopes in higher dimensions. Wikipedia contains the followin statement: In five and ...
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### Do the given perimeter and area corresponds to many shapes? [closed]

I have a perimeter P and area A of a planar shape. How to prove that there are many shapes that corresponds to those perimeter and area values?
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### In $\triangle ABC$ with $D$ on $\overline{AC}$, if $\angle CBD=2\angle ABD$ and the circumcenter lies on $\overline{BC}$, then $AD/DC\neq 1/2$

Let $\alpha$ be $\measuredangle ABD$ Let $\beta$ be $\measuredangle DBC$ Let D be a point on AC such that BD passes through the origin point O Prove that $\frac{AD}{DC}$ cannot be equal to ...
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### Euclidean geometry book for math contests

I'm a last year high school student, I'm looking for a "short" (by short I mean, not over 250 pages) Euclidean geometry book that covers topics linked to euclidean geometry of math contests, I have a ...
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### In $\Delta ABC$, find $\cot\dfrac{B}{2}.\cot\dfrac{C}{2}$ if $b+c=3a$

If in a triangle ABC, $b+c=3a$, then $\cot\dfrac{B}{2}.\cot\dfrac{C}{2}$ is equal to ? My reference gives the solution $2$, but I have no clue of where to start ? My Attempt  \cot\dfrac{B}{2}.\cot\...
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### Circles within ellipses

What is the largest circle centered at $(x_0, y_0)$ that is totally enveloped by an ellipse with a major axis $A$ and minor axis $B$? In this problem, assume a constraint s.t. the major axis is in the ...
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### 3D geometry, what are the coordinates of the 4th vertex and the point of intersection of this trapezoid?

3 Vertex of the trapezoid are given : A(4,-1,2) B(7,1,-3) D(0,-4,6) and we know that AB and CD are parallel, and CD=2AB (opposite vertices are B-D and A-C) The question is : what are the coordinates ...
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### Constant lengths (dimensional constants) and scaling

Suppose in the $xy$-plane we have defined the constant length $L$. This can be a fixed radius of a circle; or a boundary condition or any condition such, that $L$ has dimension of "meters" and is ...
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### Prove triangle area formula for barycentric coordinates

Let $P_1, P_2, P_3$ be points with barycentric coordinates (with reference triangle $ABC$) $P_i = (u_i, v_i, w_i )$ for $i = 1, 2, 3$. Then the signed area of $\Delta P_1P_2P_3$ is given by the ...
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### Prove that a circle can be inscribed iff the given condition is satisfied

I have the following question with me: "Let $B_1$ and $C_1$ be points on the sides $AC$ and $AB$ of a triangle $ABC$. Lines $BB_1$ and $CC_1$ intersect at point $D$. Prove that a circle can be ...
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### how is the following curve not simple curve? [closed]

if we start from just above point and go up and end at just below point how is the curve not simple curve. ncert class 6 chapter 4 pg 72 says it is not http://ncert.nic.in/textbook/textbook.htm?femh1=...
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### Minimum number of moves required to obtain chess like coloring.

I've been preparing for a competition and there is this problem that I cannot solve. Can you please help me and also tell me how to do similar problems if they appear in the future? Problem:
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### Defining the Cosine Function from First Principles, intuitively

Throughout most of my mathematics education, the cosine function has been defined formally either using its series expansion, as $\mathfrak{Re}(\exp i\theta)$, or as the unique solution to $y+y''=0$ ...
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### Generalized Circumcenter: minimizing the range of distances from a point to the vertices of a polygon

It is well known that the circumcenter of a polygon exists if and only if the polygon is cyclic. I would like to extend the definition of a circumcenter for noncyclic polygons. Let us define $c(A)$ ...
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### Can't figure out this triangle geometry problem

I have the following triangle: The following information about it are given: ABCD is a trapezoid (AB || DC) EF || DC Q is the intersection of AC, DB, PN, & EF Prove that EQ =...