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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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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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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 $...
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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 $...
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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 ...
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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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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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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 ...
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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 ...
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Compute $m ( \angle ACD $).

Let $\triangle ABC $ s.t $m (\angle A)=100^{°}, m (\angle B)=20^{°} $. Let $D\in Int (\triangle ABC) $ s.t. $m (\angle BAD)=30^{°} $ and $[BD $ is the bisector of $\angle B $. Compute $m ( \angle ACD ...
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Line containing centers of circles formed by intersection points of 3 circles passes through the radical center of the three circles

While messing around with the radical center of three circles, I found out the following thing: Given three circles intersecting in the points $M,N,O,P,Q,R$ as in the picture, the line joining the ...
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Compare Geometrical Shape and Subset (Transformation Invariant comparission)

I am working on CAD Automation projects, which requires comparison of two geometrical shape (Transformation invariant comparison), the expected outcome could be exact match or subset match. Please do ...
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Stuck on a Geometry Problem

$ABCD$ is a square, $E$ is a midpoint of side $BC$, points $F$ and $G$ are on the diagonal $AC$ so that $|AF|=3\ \text{cm}$, $|GC|=4\ \text{cm}$ and $\angle{FEG}=45 ^{\circ}$. Determine the length of ...
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Brocard's theorem - $O$ is orthocenter of $\Delta EFI$

Consider $ABCD$ is cyclic quadrilateral, the intersection of $AD$ and $CB$ is $E$, $AB$, $CD$ is $F$; $AC$ and $BD$ is $I$. Prove that $O$ is orthocenter of $\Delta EFI$. Firstly, i will prove $OI\...
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“Given three parallel straight lines. Construct a square three of whose vertices belong to these lines.” Are all three lines required?

Given three parallel straight lines. Construct a square three of whose vertices belong to these lines. What does "belongs" mean in the context of this question? Do the three lines have to be used ...
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Two equal segments, two known angles, find angle in triangle

This is the problem. I have tried the obvious without success. Drawing some perpendiculars (from A for example), I only know the angle $\sphericalangle ADC = 54 ^{\circ}$. How can I use the equality $...
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EGMO Problem 3.20 (BAMO 2013/3)

The problem statement follows: Let $H$ be the orthocenter of an acute angle triangle $ABC$. Consider the circumcenters of triangles $ABH$, $ BCH$, and $CAH$. Prove that they are the vertices of a ...
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Is there a possible geometric method to find length of this equilateral triangle?

Problem Given that $AD \parallel BC$, $|AB| = |AD|$, $\angle A=120^{\circ}$, $E$ is the midpoint of $AD$, point $F$ lies on $BD$, $\triangle EFC$ is a equilateral triangle and $|AB|=4$, find the ...
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Visualization of 2-dimensional projective transformation

In analytic photogrammetry, there is a transformation called 2-dimensional projective transformation which makes a link between the map plane and the picture plane. If $(X,Y)$ is a point in the map ...
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Nine-point circle - proof using plane geometry

I am taking a course in multivariable calculus this year & I thought it would be a good idea to brush up plane and solid geometry. I would like to prove that, for any given triangle, there is a ...
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Intriguing geometry problem regarding isogonal lines

A line $r$ contains the points $A,B,C,D$ in this order. Let $P\notin r$ such that $$\angle APB=\angle CPD$$ Denote furthermore by $G$ the intersection of the angle bisector of $\angle APD$ and $r$. ...
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In acute $\triangle ABC$, show $DE+DF \leq BC$, where $D$, $E$, $F$ are the feet of the altitudes from $A$, $B$, $C$, respectively. [duplicate]

Let $\triangle ABC$ be an acute angled triangle. The feet of the altitudes from $A$, $B$, and $C$ are $D$, $E$, and $F$, respectively. Prove that $$DE+DF \leq BC$$ and determine the triangles ...
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Three equilateral triangles form a hexagon [closed]

As I posted yesterday, I was learning about vectors yesterday. I know how to add and subtract them, but I can’t multiply yet. So here is an extra problem from my teacher I need help with: Given a ...
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Show that $MNPQ$ is a square

Let $ ABCD $ a quadrilateral s.t. $AC=BD $ and $m (\angle AOD)=30°$ where $O=AC\cap BD $. Let $\triangle ABM, \triangle DCN, \triangle ADN, \triangle CBQ $ equilateral triangles with $Int (\triangle ...
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What is the radius and center of the image of $|z|=1$ under $ f(z) = \frac{3z+2}{4z+3}$?

I would like to compute the image of the circle $|z|=1$ about the fractional linear transformation: $$ f(z) = \frac{3z+2}{4z+3} $$ In particular, I'd like to compute the new center and radius. ...
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1answer
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Napoleon-like theorem concerning squares erected on sides of midpoint polygon of octogon

Given an arbitrary octagon, construct it's midpoint polygon(the midpoint formed by the midpoints of the sides). Erect squares on the sides of the midpoint polygon, all inwards or all outwards. ...
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3answers
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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 =...
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High school geometry problem: Reflect a vertex about opposite side.

Let $ABC$ be a triangle and $A'$ be the reflection of $A$ about $BC$. Let $M$ be the mid-point of $BC$ and $I$ be the point of intersection of $A'M$ with the circumcircle of triangle $ABC$ (See figure)...
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Middle lines in octagon are congruent

Let $A_1A_2\dots A_8$ be a cyclic octagon. Suppose that $A_1A_6||A_2A_5, A_3A_8||A_4A_7$, and that $A_2A_5\perp A_3A_8$. Prove that the length of the two midline segmets, one connecting the midpoints ...
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1answer
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Apollonius special case.

Given circles $c$ and $d$ (green circles) and chord $AB$ arranged as shown in the figure below. How can I construct the blue circle tangent to all previous elements? [Once I think I found an ...
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Let E, F be points on the side AB of a triangle ABC. Prove that if ∠ECA = ∠BCF, then |AE||AF| / |BE||BF| = |AC|^2/|BC|^2

Let E, F be points on the side AB of a triangle ABC. Prove that if $∠ECA = ∠BCF$, then $${|AE||AF|\over |BE||BF|}={|AC|^2\over |BC|^2}$$ Please solve using power of a point. I have been trying ...
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Find angle without trigonometry

I solved the following problem using the sine law. Desired value is $\angle MAC=10°$. Can you find a geometric solution?
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Question about inversion images

"Let $W$ be a circle with center $O$ and radius $r$. Let $S$ be a point outside the circle. Let $l_1$ and $l_2$ be two tangent lines to the circle $W$ passing through the point $S$ . Let $T_1$ and $...
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1answer
154 views

Finding Angle in Triangle inside square

I need help to solve the following problem. Consider a square in the Euclidean plane with vertices $A,B,C,D\in\mathbb{R}^2$ (named in counterclockwise direction). Moreover, let $P$ be a point ...
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Angles of triangle $\triangle XYZ$ do not depend on the position of point $P$ (proof needed)

Let $ABCD$ be a fixed convex quadrilateral and $P$ be an arbitrary point. Let $S,T,U,V,K,L$ be the projections of $P$ on $AB,CD,AD,BC,AC,BD$ respectively. Let $X,Y,Z$ be the midpoints of $ST,UV,KL$. ...
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Let $ABCD$ be a rectangle, $E$ midpoint of $DC$ and $G\in AC$ such that $BG\bot AC$. Let $F$ be a midpoint of $AG$. Prove $\angle BFE =\pi/2$.

Let $ABCD$ be a rectangle, $E$ midpoint of $\overline{DC}$ and $G$ point on $\overline{AC}$ such that $\vec{BG}$ is perpendicular to $\vec{AC}$. Also, let $F$ be a midpoint of $\overline{AG}$. Prove ...
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Quadrilaterals and parallel lines: An olympiad question

Let $ABCD$ be a convex quadrilateral $\measuredangle ADC = \measuredangle BCD > 90$. Let $E$ be the point in which line $AC$ intersects the line parallel to $AD $ through $B$ and Let $F$ be the ...
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2answers
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Geometry problem solving involving alternate segment theory

In the diagram below, prove that: $$\angle QMP=\angle RMP$$ . I am pretty sure that we need to use the alternate segment theory here but I am not sure how?
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Another beauty hidden in a simple triangle (3)

In an arbitrary triangle $ABC$ pick arbitrary points $D\in BC$ and $E\in AC$ such that $DE \nparallel AB$. Denote midpoint of segment $BD$ with $F$ and midpoint of segment $AE$ with $G$. Now draw ...
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Let $ABCD$ be a parallelogram. Show that $\angle BQD = 90^ \circ$.

Let $ABCD$ be a parallelogram. There is point $P$ on the $BD$ such that $AP=BD$. $Q$ is the midpoint of the $CP$. Show that $\angle BQD = 90^ \circ$.
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Locus in the case of an ellipse.

Given that $ S $ is the focus on the positive x-axis of the equation$ \frac{x^{2}}{25} + \frac{y^{2}}{9} =1$. Let $P=(5 \cos{t}, 3 \sin {t})$ on the ellipse, $SP$ is produced to $Q$ so that $PQ = 2PS$....
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Angles inside a triangle

$P$ is a point inside triangle $ABC$ such that $\angle ABP=20^{\circ} $ , $\angle PBC=10^{\circ}$, $∠ ACP = 20°$ and $∠ PCB = 30°$. Determine $∠CAP$ , in degree. No figure was given I used the ...
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Line from incenter bisects side

I got a problem recently, and have been unable to solve it. Let $\Delta ABC$ with incenter $I$ and the incircle tangent to $BC$ at $D$. Let $M$ be the midpoint of $AD$. Prove that $MI$ bisects $BC$....
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1answer
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Making a quadrilateral cyclic by an homothety

I am wondering if it is possible to map a (convex) quadrilateral to a cyclic quadrilateral by a homothety? Or is the property of being a cyclic quadrilateral preserved by a homothehty? I would be ...
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Find $k$ (slope) if there's a homothety centered at origin with $k$ coefficient and it moves point $A(2;3)$ to point $B(2x-1;x)$

Find $k$ (slope) if there's a homothety centered at origin with $k$ coefficient and it moves point $A(2;3)$ to point $B(2x-1;x)$ First I did this $\frac{2}3=\frac{2x-1}x$ to find $x$, I got $x=\frac{...
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Use complex numbers to prove that the composition of rotations is another rotation

From ACOPS by Paul Zeitz, q4.2.30, a section on complex numbers. "Let $R_a(\theta)$ denote the transformation of the plane that rotates everything about the center point $a$ by $\theta$ radians ...
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A well-known lemma about curvilinear circles!

Let $AB$ be a chord of a circle $\Gamma$. Let $\omega$ be a circle tangent to chord $AB$ at $K$ and internally tangent to $\omega$ at $T$. Then ray $TK$ passes through the midpoint $M$ of the arc $AB$ ...
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Constructing a triangle with given $\alpha, a, m_b$.

I've to construct a triangle with given $\alpha,a,m_b$ where $m_b$ is a median to $b$. Here is my attempt. $c_1$ is a set of all possible points $A$ (e.g. we see line $BC$ at $\alpha$). $c_2$ is the ...
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1answer
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Prove that $MX=MY$.

Let $\triangle ABC$ an acute triangle and $O$ the middle of $[BC]$. Let $\mathcal{C}$ the circle with center in $O$ and radius $OA$. Let $AB\cap \mathcal{C}=\lbrace D \rbrace$, $AC\cap \mathcal {C}=...
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1answer
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Why are circles homothetic in this problem?

I'm going through Solution 1 for problem G2 from IMO 2006 (https://www.imo-official.org/problems/IMO2006SL.pdf). In the penultimate paragraph they conclude that homothety $h$ takes circle $(ABP)$ to ...
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If $|z| < 1$, prove that $\Re \left(\frac{1}{1 - z} \right) > \frac{1}{2}$.

If $|z| < 1$, prove that $\Re \left(\frac{1}{1 - z} \right) > \frac{1}{2}$. My attempt: Consider $\frac{1}{1 - z}$. Let $z = x + iy$, we know that $|z| < 1 \implies x, y < 1$. $$\frac{1}{...