Questions tagged [geometric-transformation]

A geometric transformation is any bijection of a set having some geometric structure to itself or another such set. (from Wikipedia)

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3answers
158 views

How to reconstruct a quadrilateral ABCD only using compass and straight edge?

Reconstruct a quadrilateral ABCD given length of its sides and the length of the midline between the first and third sides (namely all the segments drawn in the given figure) using compass and ...
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2answers
29 views

relative transformation of coordinates on a flat surface

I have a few coordinates that form a triangle. I have a relative point to that triangle. if the coordinates get translated to a new triangle I want to calculate the new relative point. How do I do ...
2
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2answers
156 views

Prove that the line $XY$ goes through a fixed point where $X,Y$ are on fixed conic so that $\angle XPY = 90$ where $P$ is fixed on the conic.

Say $\mathcal{C}$ is some conic and $P\in \mathcal{C}$ is fixed point on it. For each $X$ on $\mathcal{C}$ let $Y$ be such on $\mathcal{C}$ that $\angle XPY = 90^{\circ}$. Prove that the line $XY$ ...
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31 views

3D Vector projection of real and imaginary part respectively and phase difference computation

I'm a bit confused regarding the phase relationship of vector components between the real and imaginary part of a complex 3D vector. I have a complex vector of an electric field (real and imaginary ...
2
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1answer
55 views

Question about an elementary geometric construction

We have a circle $\Gamma$ and two points $A$, $A'$ on this circle. We have a line $\Delta$ and a point $P$ on this line: $P \in \Delta$. Do you know a way to construct points $M \in \Gamma$ such that ...
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2answers
104 views

Prove that H is orthocentre of ABC using inversion

Three equal circles pass through a given point $H$ and meet one another two by two at $A,B,C$ prove that $H$ is orthocentre of triangle $ABC$. My try - I proved it using elementary geometry methods ...
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2answers
43 views

Proof regarding a parallelogram and a given line segment parallel to its side

In parallelogram $ABCD$ there is given a segment $\overline{EF}$ s.t. $\overline{EF}\parallel\overline{BC}.\;$ If $G$ is the intersection point of $BE$ and $CF$ and $H$ is the intersection point of $...
3
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2answers
92 views

Olympiad math problem - Show that a pair of lines is parallel

Let $ABCD$ be a parallelogram. Draw a circunference that passes through $A$. It intersects $AB$,$AD,$ and $AC$ (for the second time), at points $E,F,$ and $G$, respectively. Lines $EG$ and $FG$ ...
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1answer
44 views

mapping low dimensional space to high dimensional space injectively?

Is it possible to find an injective mapping that transforms low dimensional space like $\mathbb{R}$ to a subset of $\mathbb{R}^2$ with interior points? For instance, can we find a mapping from $\...
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0answers
48 views

Calculating transformation matrix for perspective transform

I'm using the getPerspectiveTransform() method of OpenCV (description here). This method calculates a perspective transform from four pairs of the corresponding points. So we have to specify four ...
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4answers
491 views

Find angle in triangle $ABC$ with cevian line $AD$, such that $AB=CD$.

As you can see in the picture, there is a triangle $ABC$ with $∠C=30°$ and $∠B=40°$. Now we assuming that $AB=CD$, try to find the exact value of $∠CAD$. My attempt: Denote $∠CAD$ by $x$, we know ...
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4answers
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Finding angle $x$, a geometry question

Here is the question. Find the value of $x$. I have solved this question by my own (with 3 different methods). However, all methods of mine are based on the construction of equilateral triangles.I am ...
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4answers
660 views

For regular tetrahedron $ABCD$ with center $O$, and $\overrightarrow{NO}=-3\overrightarrow{MO}$, is $NA+NB+NC+ND\geq MA+MB+MC+MD$?

Let $ABCD$ be a regular tetrahedron with center $O.$ Consider two points $M,N,$ such that $\overrightarrow{NO}=-3\overrightarrow{MO}.$ Prove or disprove that $$NA+NB+NC+ND\geq MA+MB+MC+MD$$ I ...
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4answers
224 views

Square inscribed in a right triangle problem

Let A be a point on a fixed semicircle with diameter BC. MNPQ is a square such that $M \in AB, N \in AC, P \in BC, Q \in BC$. Let D be the intersection of BN and CM and E be the center of the square. ...
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2answers
105 views

A problem about equilateral triangle

In below equilateral triangle $ABC$, $AD=CE$, $DH=GH$. Prove that $BE=BG$ My thoughts: It looksl ike we need to prove that $ABGD$ are on the same circle but I am not sure how to use the condition $DH=...
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1answer
12 views

How the coefficient and constant change from a conic section that reflect with a line $x=3$? Where its coefficient and constant are integer.

Let M is a conic section with general equation $Ax^2+Bxy+Cy^2+Dx+Ey=F$ and assume all of the coefficient (A,B,C,D and E) and constant (F) are integer. When the conic section M reflect on a line $x=3$...
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1answer
63 views

The uniqueness of homogeneous matrices of some geometric transformations

Given the description of a specific geometric transformation, the homogeneous square matrix of it can be obtained per the methods described by many textbooks. For example, many text books suggest ...
6
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1answer
71 views

Is there exists such figure $A$ on plane $E^2$ which isometry group is isomorphic to $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$

I suppose that there are no such $A \subset E^2$ which satisfy $$\text{Iso}(A) \simeq \mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$$ But I'm stuck on showing this in formal way. Can we use ...
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4answers
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Geometry - Prove a right triangle resulting from three inscribed circles

Two half circles and a full circle fit inside a larger quarter circle as shown in the diagram. The centers of the two half circles are on the two sides of the quarter circle, respectively. Prove that ...
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1answer
99 views

Coordinate axes transformation

I'm trying to transform from regular math coordinates to computer pixel coordinates as shown. 1) Each unit on left system equals $20$ units in right system. 2) On left system, the origin is at ...
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2answers
67 views

Four touching circles and one common tangent

Given four touching circles and one common tangent, show that $$\angle BAD = \dfrac{1}{2}(\angle DO_1A + \angle AO_2B)$$ It is done by looking at triangles $\triangle ADO_1$ and $\triangle ABO_2$. ...
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1answer
33 views

Transformation question confusion

Let me ask again, as clearly my previous post people did not understand what I was asking. Just to clarify this is a middle school question which kind of made me forget alittle on transformations. ...
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2answers
31 views

$AD, BE, CF$ are altitudes in $ABC$. $P,Q$ are points on lines $BC$ and $AB$ so $QP =PF$ and $R$ on $AC$ so $RP =CP$. Prove $ QDRA$ is a cyclic

Let $AD, BE, CF$ be the altitudes of triangle $ABC$ and $P$ be an arbitrary point of side $BC$. Point $Q$ on the line $AB$ is such that $QP =PF$ and the point $R$ on the ...
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2answers
152 views

Olympiad Geometry | Homothety 2

Two noncongruent circles intersect at $X$ and $Y$. Their common (external) tangents intersect at $Z$. One of the common tangents touches the circles at $P$ and $Q$. Prove that $ZX$ is tangent to the ...
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3answers
83 views

Given equilateral $\triangle ABC$ and $M$ at distances $3$, $5$, $4$ from $A$, $B$, $C$, find $\angle AMC$.

Given that $\triangle ABC$ is equilateral triangle, and $M$ is a point inside $\triangle ABC$ such that $$AM=3\;\text{units} \qquad BM=5\;\text{units} \qquad CM=4\;\text{units}$$ what is the ...
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1answer
76 views

In a $\Delta ABC$ prove: $\angle AST=\angle OSB$ if $T$ is the midpoint of $AH$ and $S=BC \cap t$ ($t$ = tangent line in $A$ to a circumcircle)

Let $ABC$ a triangle and $H$ and $O$ its orthocenter and circumcenter. Denote by $T$ the midpoint of the segment $AH$ and by $S$ the intersection of the tangent in $A$ to the circle $ABC$ and the line ...
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1answer
49 views

Two circles meet at $A$ (and $B$), and their common tangents meet at $O$. Show that $\overline{OA}$ bisects the angle made by the two tangents at $A$. [closed]

Two circles are cut at $A$ and $B$, and their common tangents meet at $O$. If $AP$ and $AQ$ are the tangents at $A$ to the circles, how do you prove that $OA$ bisects $\angle PAQ$? I have tried using ...
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3answers
119 views

Finding angle in the isosceles triangle [closed]

There is the following triangle. I infer that the triangle is isosceles. But I cannot go further.
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0answers
77 views

Let $ ABC$ be a triangle with circumcentre $ O$…

Let $ ABC$ be a triangle with circumcentre $ O$. The points $ P$ and $ Q$ are interior points of the sides $ CA$ and $ AB$ respectively. Let $ K,L$ and $ M$ be the midpoints of the segments $ BP,CQ$ ...
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2answers
53 views

Geometry transformation problem

The question is :- A figure consist of five equal squares in the form of a cross .show how to divide it by two straight cuts into four equal figures which will fit together to form a square. ...
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1answer
38 views

A $\triangle {ABC}$ is rotated in its own plane about point $A$ into position $A'B'C'$.if $AC$ bisects $BB'$ prove that $AB'$ bisects $CC'$

The question is :- A $\triangle {ABC}$ is rotated in its own plane about point $A$ into position $A'B'C'$.if $AC$ bisects $BB'$ prove that $AB'$ bisects $CC'$ To be clear the main problem is I ...
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1answer
44 views

Angle bisector and perpendicular lines

$\angle POQ$ with angle bisector $OL$ is given. On its rays, $OP$ and $OQ$, points $A$ and $B$ are chosen, such that $OA < OB$. $a$ is the perpendicular line through $A$ to $OP$, and $a$ intersects ...
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1answer
36 views

Symmetrical points

$\triangle ABC$ is given, in which $AC > BC$ and the incircle $k(O)$ touches $BC$ and $AC$ in $M$ and $N$, respectively. Point $B_1$ is the image of B with respect to the line $CO$. Show that $M$ ...
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3answers
61 views

Be $E,F,K , L,$ points in the sides $AB,BC,CD,DA$ of a square $ABCD$, respectively. Show that if $EK$ $\perp$ $FL$ , then $EK=FL$

Be $E,F,K , L,$ points in the sides $AB,BC,CD,DA$ of a square $ABCD$, respectively. Show that if $EK$ $\perp$ $FL$ , then $EK=FL$. I need help proving something like this: Any hints? Edited: I ...
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2answers
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Prove that ED=EF [closed]

In the diagram below $AD=DC$ and $AE=EB$ and both triangles $AEB$ and $ADC$ are right angel triangles and $M$ is the midpoint of $BC$ also $MD=MF$. Prove that $ED=EF$
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2answers
124 views

How to test whether two triangles can be made to coincide by rotation and translation only

I am interested in conditions that determine if two triangles can be made to coincide with each other by rotation and translation only (without mirroring). So, given two triangles, one should be able ...
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2answers
1k views

Rotate and scale a point around different origins

I am trying to rotate an arbitrary, 2D point (x,y) around another point (a,b), and at the same time, scale it from a different point (c,d). To transform the point, I must use a 3x3 transformation ...
1
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2answers
63 views

which formula could be used to get a set of vertices after rotate a triangle?

there are 3 vertices to define a triangle. print(x,y) [1 2 3] [ 2 3 -2] python could be used to plot this triangle ...
2
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1answer
96 views

Eigenvector of two rotation matrices

I am having difficulty in understanding a geometry problem which contains geometric-transformation, rotation and reflection. Background In this image, a camera with camera center $O_c$ is presented. ...
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2answers
90 views

Transformation matrix to align an object towards a vector.

I have an object(say, a prism) in 3D space and a vector. I need to align this object in the direction of the given vector. What would be the transformation matrix required to achieve it? Assume that ...
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2answers
82 views

Trouble in mapping of möbius transformation

Question:- Show that the transformation $$ w = \frac{2z+3}{z-4}$$ maps the circle $x^2+y^2-4x=0$ onto the straight line $4u+3=0$ My attempt:- The circle $x^2+y^2-4x=0$ is $|z-2|=2$ . . .$(1)$ So ...
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5answers
1k views

Geometry - Proving a common centroid.

Triangle PQR is drawn. Through it's vertices are lines drawn which are parallel to the opposite sides of the triangle. The new triangle formed is ABC. Prove that these two triangles have a ...
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3answers
86 views

Equilateral $\triangle ABC$ has $A$ fixed and B moving in a given straight line. Find the locus of $C$.

$\triangle ABC$ is an equilateral triangle with vertex $A$ fixed and $B$ moving in a given straight line. Find the locus of $C$. I think the locus of $C$ should also be a line. What I have done: ...
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1answer
50 views

Let $O$ and $O'$ be two circles intersecting at two points, $A$ and $B$. Construct a line segment going through $A$ bisected by the two circles

Let $O$ and $O'$ be two circles intersecting at two points, $A$ and $B$. Construct a line segment $l$ going through $A$ and such that if $l$ meet $O$ at $M$ and $O'$ at $M'$, the length of $AM$ and $...
2
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2answers
38 views

Proof that the triangle formed by mass centers is equilateral.

Let $ABC$ be a triangle and consider $A_1$, $B_1$, $C_1$ outside the triangle such that triangles $ABC_1$, $BCA_1$ and $ACB_1$ are equilaterals. Consider now $A_2$, $B_2$ and $C_2$ mass centres of $...
3
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1answer
72 views

Euclid Geometry: Seeking for a simpler geometric solution

The problem: Answer: Extend $AC$ to the side of $C$, such that $BC=CE$. Then, construct the equilateral triangle $BEF$. Triangles $CAB$ and $EFC$ are congruent, thus $AC=BE$. Now, observe ...
3
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3answers
52 views

Prove of a theorem of a geometrical place

I am having issues to prove the back of this theorem: Let $ABC$ be a triangle and fixed $D∈AB$. The Geometric Place of the $X$-points that form with $D$ and an arbitrary point $S∈AC$ an ...
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2answers
136 views

Prove that $BC$, $B_1C_1$, $B_2C_2$ are concurrent.

Consider altitude $AH$ of $\Delta ABC$. $B_1$ and $B_2$ are points on side $AB$ such that $HB_1 \perp AB$ and $HB_2 \parallel AC$. $C_1$ and $C_2$ are points on side $AC$ such that $HC_1 \perp AC$ and ...
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1answer
41 views

Edges of rectangle mapped to a new plane

If a rectangular plane lies in quadrant I and is transformed by a polynomial function, will the edges of the rectangle always map to the edges of the transformed plane? Will all points along the ...
5
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3answers
151 views

Circle problem in which i wanted to find value of $PQ$

Find a value of $PQ$: Let the radius of bigger circle be $R$, and that of the smaller circles be $r_1$ and $r_2$. Hence we can find value of $R = 10$. I don't know how to proceed further to find ...