Questions tagged [inversive-geometry]
Questions related to Inversive Geometry and its applications.
120 questions
1
vote
0
answers
50
views
Condition for two tetrahedra to be related by inversion
Given two labeled tetrahedra
$$
V_0V_1V_2V_3 \quad\text{and}\quad V_0'V_1'V_2'V_3',
$$
define their opposite-edge pairs
$$
(01,23),\quad (02,13),\quad (03,12).
$$
Let
$$
m_{ij}=\frac{|V_i'V_j'|}{|...
- 695
4
votes
1
answer
108
views
Prove the existence of an inversion that sends three mutually tangent circles to three equal ones
Three tangent circles contain three tangent circles; all mutual tangencies as in the labels. None of the $12$ tangent points coincide. Prove that $C,E,K,G$ are concyclic.
Cite https://en.wikipedia.org/...
- 184
1
vote
1
answer
109
views
Inversive geometry problem
Let $\Omega$ be a circle with center $O$, let $A$ and $B$ be two points on a diameter $XY$ of $\Omega$. Let $X_1$ and $Y_1$ be the points of intersection, on opposite sides of the line $AB$, of the ...
- 88
1
vote
1
answer
158
views
$P, Q, N, M$ are concyclic
Let $ABC$ be an acute triangle. The points $B, D, E, C$ are collinear in this order, and satisfy the relation $BD = DE = EC$. Let $M$ and $N$ be the midpoints of segments $AD$ and $AE$, respectively. ...
- 597
1
vote
1
answer
62
views
How to construct the center of inversion that maps a harmonic quadrilateral to a square?
An inversion at $P$ maps the quadrilateral $ABCD$ to a square $A'B'C'D'$ then $ABCD$ is a harmonic quadrilateral.
Proof: Since $\triangle A B P \backsim \triangle B^{\prime} A^{\prime} P $ we have $\...
- 6,019
1
vote
1
answer
62
views
Radius relation of four consecutively touching circles $O_7,O_2,O_3,O_6$ with common tangent circles $O_1,O_4$
A construction of four consecutively touching circles $O_7,O_2,O_3,O_6$ that all touches circles $O_1,O_4$:
Triangle $O_1O_2O_3$ has outer Soddy circle $O_4$, so that the circles $O_1,O_2,O_3,O_4$ are ...
- 6,019
4
votes
3
answers
224
views
$M$ in triangle $ABC$ ; images $A',B',C'$ by inversions wrt circles $(MBC),(AMC),(ABM)$ are such that $A'B'C'$ equilateral : is $M$ a known center?
Yesterday, a question occurred to me, and I tried to find the center of the triangle that satisfies the condition I am thinking of, but I have not been able to do so yet
Suppose a triangle $∆ABC$, ...
- 4,451
1
vote
2
answers
278
views
Pappus chain - triangle of tangency points
$A,B,C$ are collinear points. Starting with 3 circles with diameter $AB,AC,BC$, there is a chain of tangent circles, all tangent to one of the two small interior circles and to the large exterior one.
...
- 6,019
3
votes
0
answers
78
views
Geometric proof of $|\frac{e^{j\theta} - a}{1 - ae^{j\theta}}| = 1$
This can be proven algebraically,
$$\left|\frac{-a+e^{-j \theta}}{1-a e^{-j \theta}}\right|=\sqrt{\frac{(-a+\cos (-\theta))^2+(\sin (-\theta))^2}{(1-a \cos (-\theta))^2+(a \sin (-\theta))^2}}=\sqrt{\...
- 143
0
votes
0
answers
70
views
Inversive product of tow ultraparallel geodesics in the hyperbolic plane is $\cosh{\rho}$
This is Lemma 7.17.3 in Beardon:
Lemma 7.17.3 Let $L$ and $L'$ be geodesics in the hyperbolic plane. Then the inversive product $(L,L')$ is $\cosh{\rho(L,L')},1,\cos{\phi}$ according as $L,L'$ are ...
- 893
1
vote
0
answers
155
views
Invariance of the Poisson integral under inversions
Let us denote the Poisson kernel on $B_r$ by $$ P(x,\zeta)=\frac{r^2-\vert x \vert^2}{r\omega_{n-1}\vert x-\zeta \vert^n}$$ where $x\in B_r$ and $\zeta\in\partial B_r$. Given a boundary function $f$ ...
- 1,371
2
votes
1
answer
107
views
Lie Sphere Geometry, but with continuously oriented cycles
In Lie Sphere Geometry, an oriented cycle is either:
a point
a non-point circle, paired with a value in $\{-1,+1\}$ called its orientation
a line, paired with a value in $\{-1,+1\}$ called its ...
- 8,405
1
vote
1
answer
284
views
Given a circle C on the plane, what are the circles orthogonal to the x-axis and tangent to C?
Let $C$ be a generalised circle in $\mathbb{CP}^1$ represented by a $2 \times 2$ indefinite Hermitian matrix also called $C$. (See here for how an indefinite Hermitian matrix determines a generalised ...
- 8,405
1
vote
1
answer
58
views
For the reflection $\phi$ in the sphere $S(a,r)$ it holds that $\phi(B^n) = B^n$ if and only if $\phi(a*) = 0$
This is a question about Theorem 3.4.2 in "Geometry of discrete groups" from Beardon.
Let $\phi$ be the reflection in a sphere $S(a,r)$ for $a \in \hat{\mathbb{R}^n}$, $r \in \mathbb{R}$. ...
- 599
0
votes
1
answer
137
views
Prove that line is orthogonal using inversion
P' is the inverse of P with respect to the circle c and M is a point of a circle c. Line through M and P intersects with c at A and line through M and P' intersects with c at B. Prove that AB is ...
- 183
3
votes
3
answers
182
views
Prove that the tangents at $X$ and $Y$ meet on the line $AB$ - circle inversion
Let $P$ be a point outside a circle and let the tangents from $P$ touch the circle at $A$ and $B$. A line through $P$ intersects the circle in points $X$ and $Y$. Prove that the tangents at $X$ and $Y$...
- 409
0
votes
1
answer
220
views
Circle of Apollonius modified to inversion form
Let $A$ be any point outside the circle $\omega$, let $A^*$ be its inverse around $\omega$, and $P$ is any variable point on $\omega$. Prove that the ratio $PA$/$PA^*$ is constant, and hence find its ...
1
vote
0
answers
127
views
The image of a line through an inversion
I am going to start by saying that geometry is not my strong suit, but I am taking a course on analytic geometry where I learnt about inversions and there is this question that bugs me.
The following ...
- 2,867
0
votes
1
answer
39
views
expressing class of inversions in einstein summation convention
I'm working on a personal project where I'm studying the set of inversions in $\Bbb R^n$
that preserve the unit sphere centered at the origin; these transformations can be defined based on a center of ...
- 344
0
votes
1
answer
115
views
Which property of polar has been applied to this proof.
Which property of polar has been used please give its proof also.
- 81
4
votes
1
answer
315
views
Show that $A-$excircle is tangent to $(AST)$
Elmo is now learning olympiad geometry. In triangle $ABC$ with $AB\neq AC$, let its incircle be tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The internal angle bisector of ...
- 4,265
3
votes
1
answer
238
views
Doubt: Prove that the circumcircles of $\Delta ABC$ and $\Delta ADE$ are tangent with $\sqrt {BC}$
So, I recently started inversion and I have doubt in this solution . It's from "A beautiful Journey through Olympiad geometry " by Stefan Lozanovski. This Problem uses $\sqrt{BC}$
Here , I ...
- 4,265
5
votes
4
answers
2k
views
Find a circle perpendicular to two other circles.
Here are two circles $C_1 = [ x^2 + y^2 = 1]$ and $C_2 = [ (x-2)^2 + y^2 = 3 ]$. The radii are $1$ and $\sqrt3$ and the two circles are orthogonal to each other $C_1 \cdot C_2 = 0$. What is the ...
- 25.2k
0
votes
0
answers
71
views
How to determine if three distinct points $a,b,c \in \Bbb c$ are collinear using Mobius Transformation?
Given three points $\frac{3}{2} + i , 2i,-6+6i$. I have the mobius transformation that maps these three points to $0,1,\infty$ respectively as
$M(z) = \frac{(-4i+6)(z-(1+2i))}{(3-7i)(z-(10-20i)}$
...
- 303
0
votes
1
answer
1k
views
Inversion in circle
Let $C$ be a circle with the middle point $O$ and the radius $r$, we say that the points $P$ and $P'$ are inverse points with respect to $C$ if:
$1.$ $|OP|·|OP'|=r^2$
$2.$ $P$ and $P'$ are on the ...
