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

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4
votes
2answers
52 views

Angle chase:In $\Delta ABC, AB=AC $ and $\angle BAC=20°.$ If $CD$ is the median from $C$ to side $AB$, find $\angle ADC$.

In $\triangle ABC, AB=AC $ and $\angle BAC=20^\circ$ If $CD$ is the median from $C$ to side $AB$, find $\angle ADC$.
3
votes
0answers
38 views

Area of $A'B'C'$ is to area of $ABC$ is $\frac{(m-n)^2}{m^2+mn+n^2}$

In the sides $BC,CA,AB$ are taken three points $A',B',C'$ such that $BA':A'C=CB':B'A=AC':C'B=m:n$.Prove that if $AA',BB',CC'$ are joined they will form by their intersections a triangle whose area is ...
1
vote
1answer
33 views

Prove that the length of the common chord is $\frac{2ab\sin \theta}{\sqrt{a^2+b^2+2ab\cos \theta}}$

Two circles ,of radii $a$ and $b$,cut each other at an angle $\theta.$Prove that the length of the common chord is $\frac{2ab\sin \theta}{\sqrt{a^2+b^2+2ab\cos \theta}}$ Let the center of two circles ...
-5
votes
1answer
43 views

Geometry - Trapezium and its properties

In the trapezium $ABCD$ , $AB$ is parallel to $CD$ and $O$ is the intersection of $AD$ and $BC$. The line $PS$ is drawn through $O$ in such a way that $PS$ is parallel to $DC$. If $AB = 20$ AND $CD = ...
3
votes
1answer
43 views

Prove that the ratio of the areas of the triangles $A'B'C'$ and $ABC$ is $2\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$

If the bisectors of the angles of a triangle $ABC$ meet the opposite sides in $A',B',C'$,prove that the ratio of the areas of the triangles $A'B'C'$ and $ABC$ is $2\sin \frac{A}{2}\sin \frac{B}{2}\sin ...
1
vote
2answers
30 views

Geometric Interpretation of Determinant of an Inverse Matrix

The $\mathbf{A}$ be an $n\times n$ full rank matrix. Then, the (signed) volume enclosed by the rows (or columns) of $\mathbf{A}$ is equal to $\det(\mathbf{A})$. My question is, what is a geometric ...
2
votes
3answers
40 views

Triangle area inequalities

I've got stuck on this problem : Proof that for every triangle of sides $a$, $b$ and $c$ and area $S$, the following inequalities are true : $4S \le a^2 + b^2$ $4S \le b^2 + c^2$ ...
2
votes
1answer
32 views

How is this circle inversion formula calculated?

I know about the inversion of a point inside a circle. But I was reading Peter Sarnak's paper on the Apollonian gasket, and got to the part where he was trying to prove descartes circle theorem. He ...
4
votes
1answer
44 views

Geometry of Analysis

I am a recently graduated student and doing Post Graduation now. I often come across uniform convergence, uniform continuity etc. As we all know that we check continuity and convergence easily by just ...
0
votes
1answer
41 views

Triangle, vectors, hard to explain

P is the middle of a median line from vertex A, of ABC triangle. If Q is the point of intersection of lines AC and BP. Find relations of $|\vec{AQ}|$/$|\vec{QC}|$ and $|\vec{BP}|$/$|\vec{PQ}|$ Any ...
0
votes
0answers
8 views

3D point rotation round a fix reference point

I want to compute a transformation from 3D point A to 3D point B through a reference point 0 which is fixed. I have the 6DOF transformation from A - 0 and B - 0. That is x,y,z and Quaternions of ...
2
votes
4answers
70 views

show that out of all triangles inscribed in a circle the one with maximum area is equilateral

show that out of all triangles inscribed in a circle the one with maximum area is equilateral How do i start. I have to use function of two variables Thanks
0
votes
0answers
25 views

Simple problem of three similar tirangles

Take $ABC$ a triangle sucht that $AC=CB$. Let $O$ the middle point of the base $AB$ and $\gamma$ the circle centered in $O$ with tangent points $P$ and $Q$ on sides $CB$ and $AC$. Draw a line tangent ...
-1
votes
2answers
23 views

to find the equation of line when distance from cordinates is given [closed]

Through a point $P(4,1)$ a line is drawn to meet $3x-y=0$ at $Q$ where $PQ=\frac{11}{2\sqrt2}$. Determine the equation of the line.
4
votes
6answers
213 views

$\arctan (x) + \arctan(1/x) = \frac{\pi}{2}$ [duplicate]

How can I show that $\arctan (x) + \arctan(1/x) =\frac{\pi}{2}$? I tried to let $x = \tan(u)$. Then $$ \arctan(\tan(u)) + \arctan(\tan(\frac{\pi}{2} - x)) = \frac{\pi}{2}$$ but it does not ...
3
votes
0answers
16 views

what allows certain polygons(not necessarily regular) to tessellate?

there are irregular pentagons that tessellate. Is there some observable property? please provide anything else you want to relative to the question, thank you!
1
vote
1answer
41 views

Reducing the perimeter of a plane without reducing area??

There is rectangle pool with an area of 36 square-yards, then there is another pool of the same area, but with a smaller perimeter. How is this achieved?
0
votes
0answers
7 views

Cross-verifying a homography on known correspondences

Context I have two sets of known 2D correspondences $S_1$ and $S_2$, from which I have constructed homographies $h_1$ and $h_2$. This was achieved using the homogeneous estimation method, ie. by ...
1
vote
1answer
33 views

$S \leq \frac{(a+b)(c+d)}4 $

I got stuck on this problem: Given a convex quadrilateral of area $S$ and sides $a$, $b$, $c$ and $d$, prove that: $$S \leq \frac{(a+b)(c+d)}4$$ What I've done so far was to proof that ...
1
vote
2answers
46 views

What is the general equation equation for rotated ellipsoid?

I have general equation for ellipsoid not in center: $$ \frac{(x-x_0)^2}{a^2}+\frac{(y-y_0)^2}{b^2}+\frac{(z-z_0)^2}{c^2}=1.$$ What is the equation when it's rotated based on $\alpha$(over $x$ axis), ...
6
votes
0answers
150 views

Why Newton wanted lines to be generated by continued motion of points rather than by apposition of parts? [migrated]

The following passage has been extracted from the Newton's (John Stewart's English translated version) "Sir Issac Newton's two Treatises: Of the Quadrature of Curves, and Analysis by equations of an ...
1
vote
0answers
17 views

Derivation/equation for solid angle factor correction

Derivation/equation for solid angle factor correction Summary: I want to determine a correction for the Solid Angle Factor (SAF) due to partially overlapping 'outer' spheres (of different sizes), as ...
0
votes
1answer
37 views

simple exercise of euclidean geometry

I've to solve this simple exercise but i can't see how. Problem: let $AB$ and $CD$ two equivalent ropes of one circle of centre $O$. Let $P$ and $Q$ two points that belong on the extentions of the ...
4
votes
2answers
70 views

Prove that having 6 points in the interior of a square

Prove that having 6 points in the interior of a square of side length 3, we can choose 2 of them so that the distance between them is less than 2. Looks obvious, but I can't get a rigorous ...
1
vote
1answer
136 views

Parallelizability of Lie groups

I have tried a lot to prove this well-known result. The basic idea behind the proof is clear to me. But I'm stuck at showing either of the following conditions: The map $\ G \rightarrow TG \ $ given ...
0
votes
1answer
42 views

Prove that 5 points lie on the same circle

How do I prove that 5 points lie on the same circle? I know about the theorem that opposite angles in a quadrilateral are supplementary, but how does that help me prove that 5 points lie on the same ...
0
votes
0answers
20 views

Proof of Menelaus using areas

I've tried to proof Menelaus' theorem using areas, but I've didn't figure out how. Some suggestions would be appreciated. Menelaus' Theorem states : Given a triangle ABC and a transversal ...
1
vote
1answer
15 views

Proof a special extension on the isoperimetric theorem

Let both ends of a string of length $L$ be tied to a stick of length $S$. Among all plane regions enclosed by this contraption, it achieves maximum when the string forms a circular arc. It is noted ...
0
votes
1answer
30 views

Equation of the plane tangent to the given area

Find the equation of the plane tangent to the surface: $$x^{\frac{1}{3}}+y^{\frac{1}{3}}+z^{\frac{1}{3}}=1$$ at the point: $$P=\left(1,-1,1\right)$$ How to find it? I know i have to calculate a ...
2
votes
3answers
40 views

Proving a ratio that has a relation with the Perpendicular bisectors and circumcircle

$ABC$ is a triangle, $D$ is a point on the side $BC$ of $\triangle ABC$, $R_b$ is circumradius of $\triangle ABD$ , and $R_c$ is the circumradius of $\triangle ACD$. Prove that $$ {Rb\over Rc} ...
3
votes
1answer
57 views

High School Geometry Text?

This year I will be teaching 8 hard-working home-educated teens a Geometry course. Back in 1994-1999 I worked full time as a High School educator, taking a turn teaching everything from Pre Algebra ...
2
votes
2answers
32 views

Proof of correctness of a formula for the area of a polygon

Let $P$ be a $n$-gon with vertices $(x_1,y_1),\ldots,(x_n,y_n)$ enumerated clockwise. Then the area $\text{Area}(P)$ of $P$ is $$ \text{Area}(P) = \sum_{i=1}^n\frac{1}{2}(x_{i+1}-x_i)(y_{i+1}+y_i).$$ ...
1
vote
1answer
40 views

Prove by vector method that $p_1+p_2=p_3$

Let $ABC$ be an acute angled triangle whose incenter and centroid are respectively $I$ and $G$.$AI,BI$ and $CI$ cuts the sides of the triangle at $P,Q,R$ respectively.If $p_1,p_2$ and $p_3$ are the ...
4
votes
3answers
405 views

Find the longest side of the triangle.

The sides $a,b,c$ of a $\triangle ABC$ are in $GP$ whose common ratio is $\frac{2}{3}$ and the circumradius of the triangle is $6\sqrt{\frac{7}{209}}$.Find the longest side of the triangle. I used ...
2
votes
2answers
41 views

A circle of radius $r$ is dropped into the parabola $y=x^{2}$. Find the largest $r$ so the circle will touch the vertex.

If $r$ is too large, the circle will not fall to the bottom, if $r$ is sufficiently small, the circle will touch the parabola at its vertex $(0, 0)$. Find the largest value of $r$ s.t. the circle will ...
72
votes
9answers
3k views

Prove the theorem on analytic geometry in the picture.

I discovered this elegant theorem in my facebook feed. Does anyone have any idea how to prove? Formulations of this theorem can be found in the answers and the comments. You are welcome to join in ...
0
votes
0answers
53 views

Property of Circle Tangent and Intercepts

Is there a proof for Inverse theorem of following well known property of circles among Euclid's Elements? A line $OQP$ rotates about fixed point $O$ such that $ ( OP\cdot OQ )$ is a constant. Show ...
0
votes
1answer
20 views

Find diffeomorphism transforming the following areas:

Find diffeomorphism transforming the following: interior of the triangle T with vertices in $(0,0),(0,1),(1,0)$ onto the interior of the circle of radius 1 and centre in $(0,0)$. Obviously i am ...
0
votes
1answer
69 views

Triangle construction procedure

Two lines $L1,L_2$ pass through a common point $O. $ $L_2$ goes through points $P$ and $Q$. How to construct a circle through $P,Q$ to be tangent to $L_1?$ In a particular case, at the tangent ...
2
votes
2answers
28 views

Why is the Apollonian Gasket composed of infinitely many circles?

This famously known fractal has infinitely many circles, however I find it hard to find a rigid proof that confirms how or why this fractal is composed of infinitely many circles (and only circles). ...
1
vote
1answer
23 views

Extending flange on bent bar so horizontal tangent moves to correct place

I'm bending a bar as in the image below, but i need to extend the left flange with x to get the top of the bend (green tangent at the bend) to move up to the other green line. v, u, a, b, d and r is ...
2
votes
2answers
43 views

Quadrilateral's area problem

I have some troubles with this problem : Let $ABCD$ be a convex quadrilateral. $M$, $N$, $P$ and $Q$ are the midpoints of the sides $AB$, $BC$, $CD$ and $AD$. $AN$, $BP$, $MD$ and $CQ$ are ...
1
vote
1answer
17 views

Distance between $2$ skew lines

Suppose that $A(0,0,0), B(1,2,0), C(0,-3,2)$ and $D(3,-4,5)$ and $AB, AC$ and $AD$ are three edges of a parallelepiped. If $l_1$ is the line passing through $A$ and $B$ and $l_2$ is the line passing ...
0
votes
1answer
79 views

Geometry - angle bisector, circumcircle: SL olympiad

I tried this problem as much as I can, but I got nothing. This is a Sri Lankan mathematical olympiad problem. Let $P$,$Q$ be points on the sides $AB$ and $AC$, respectively, of a $\triangle ABC$ ...
2
votes
0answers
26 views

Collinearity problem (Newton-Gauss line)

I had some troubles with this problem : Let $ABCD$ be a convex quadrilateral. $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$. The sides $AB$ and $CD$ are extended until they ...
3
votes
3answers
42 views

$\cos \alpha+\cos(\alpha+\beta)+\cos(\alpha+2\beta)+…+\cos(\alpha+(n-1)\beta)=0 $

If each side of a regular polygon of $n$ sides subtend an angle $\alpha$ at the center of the polygon and each exterior angle of the polygon is $\beta$,then prove that $\cos ...
0
votes
0answers
16 views

Locus of points on a rotating line ; points differently ordered

A line rotates about a fixed point $O$ with ordered points $P,O,M $, while $ M $ is moving along this line $POM$. Find locus of points $ P ,M $ if $ MP^2- OM^2 = T^2 $ constant for all inclinations ...
2
votes
0answers
49 views

Intersecting convex figures

Take in the real plane a finitely long horizontal line segment and connect the two endpoints by a convex path, above the segment, with the property that the only extreme points of the convex hull of ...
0
votes
2answers
12 views

Rectangle Zoom-in: Accounting for proportions and distances between them

Let's say I have n rectangles: each with their own height and width, and each with their own coordinate on a plane. I can scale the width and height of the rectangles by let's say...S. How do I ...
2
votes
2answers
27 views

About finding maximum area [closed]

What is the maximum area of a triangle if two vertices are given?as I need to find the no. Of possibilities of third vertices ? I will then find the possibilities of third vertex using given area if ...