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List of geometric theorems linked by two squares
For now, I will only post pictures of the theorems. Perhaps later I will add links that talk about the theorems in detail
Potema's theorem:
Two squares created on the sides of a triangle $∆ABC$ And ...
- 2,296
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Maximizing area of the triangle in a quarter circle
The area of the triangle ABC is equal to its side AC multiplied by the height, divided by $2$. The height will be maximal if we draw a radius OB such that OB is perpendicular to AC. Let H be the ...
- 4,451
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Euclidean Geometry problem regarding the line passing through incenter
In fact, your existing work almost solves the problem already. The next step is to observe that the altitude $h_A$ from $A$ to $BC = x+y$ must satisfy
$$5 \sqrt{2} = |\triangle ABC| = \frac{1}{2} h_A ...
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Geometry problem with external tangent of two circles
Let us use inversion to show the following lemma, equivalent to the given problem:
Lemma: Consider two circles $\odot(O)$, $\odot(O')$, intersecting in two points, $A,B$, and having $PQ$ as common ...
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Is my trigonometric proof of the Pythagorean theorem correct?
The identity $\sin^2\theta + \cos^2\theta = 1$ is essentially the Pythagorean Theorem in the form of trigonometry.
In your proof you use the fact that triangles with congruent angles are similar.
This ...
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4
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List of geometric theorems linked by two squares
Euclid Book II Prop 6 on what we would now describe algebraically as the difference of squares:
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Geometry problem with external tangent of two circles
Let O and W be the centers of $C_3$ and $C_2$, respectively. Notice that Y, O, W are collinear by the property of tangent circles. We'll mark $∠OZX = ∠OXZ = \alpha$ and $∠OZY = ∠OYZ = \beta$. ($OZ = ...
- 105
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Maximizing area of the triangle in a quarter circle
Just extend the circle, call $A'$ and $C'$ the intersection of the line AC with the circle. The middle $B$ of the arc $A'C'$ is the solution.
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Proving that the cone on a finite Euclidean simplicial complex is again a Euclidean simplicial complex.
This is proved in Munkres's Elements of Algebraic Topology book. Essentially, every point of $w\ast\sigma$ is on a unique line segment from $w$ to some point of $\sigma$. Thus a point in the ...
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Conjecture: The line joining the incenter and the circumcenter always subtends an obtuse angle at centroid
Denote the incenter by $I$, circumcenter by $O$, the centroid by $G$. Assume that the triangle is not equilateral, so that the points $I, G, O$ are distinct. In order to prove that $\angle IGO>90^\...
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Why is maximum number of joints of 6 lines is 4?
Note that $2$ intersecting lines define a plane. Thus, a joint is formed by $3$ intersecting planes. Let's add one more plane to these three. Note that $4$ planes have at most $6$ distinct lines as ...
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Why is maximum number of joints of 6 lines is 4?
Let $G = (V,E)$ be a graph, and we're interested in the vertices which fulfill $deg(v) \geq 3$. We notice that $|E| = 6$ from the assignment, and we can use your earlier fact that a tetrahedron is a ...
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Distance between triangle incenter and vertices
I'm using $s$ for semiperimeter (that's more common).
First, drop perpendiculars from $I$ onto $a,b,c$ and call them $ID,IE,IF$ respectively. All are equal to the inradius $r$.
Then, it is well known ...
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List of geometric theorems linked by two squares
Here are somewhat-natural generalizations of a couple of the two-squares-joined-at-a-vertex results from OP's answer.
Potema's Theorem
Let squares $\square A'B'C'D'$ and $\square A''B''C''D''$ be as ...
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3
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A triangle whose area is four times the area of Napoleon's triangle
The circumcircles of the outer equilateral triangles, centred at $L$, $M$, $N$, meet at $F$, the Fermat point of triangle $ABC$. Note that $LM$ is the perpendicular bisector of common chord $CF$, $MN$ ...
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Euclidean Geometry problem regarding the line passing through incenter
You've made a good start. Using your diagram, we have that
$$2\alpha + 2\beta + 2\gamma = 180^{\circ} \;\to\; \alpha + \beta + \gamma = 90^{\circ} \tag{1}\label{eq1A}$$
Thus, using your $x + y = 3\...
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What is the $\inf$ and $\sup$ of the area of quadrilateral given its sides length?
Areas can get arbitrarily small under some circumstances, suach as edges (in order) $a,b,a,b$ which has maximal area as a rectangle, but can be squashed down as a parallelogram.
The example that ...
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What is the $\inf$ and $\sup$ of the area of quadrilateral given its sides length?
The supremum will be when the four nodes lie exactly on a circle: a cyclic quadrilateral. The area of a quadrilateral inscribed in a circle is given by Bretschneider’s formula (actually Brahmagupta's ...
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Is my trigonometric proof of the Pythagorean theorem correct?
It is neither right nor wrong , though it is useful to improve basic intuition.
Euclidean Geometry ( with the list of Axioms & Postulates including the Parallel Postulate ) gives the Pythagorean ...
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Generalizing the property of two parabolas tangent to a circle, each of which touches it at two points
Still not an answer, but I would like to share a generic SAGE program (SAGE is a free Python-like online resource ; see below this program and the way to run it). Writing this program has in fact be ...
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How many intersections are there for two confocal parabolas with their axis perpendicular to each other?
The polar equation of a parabola opening upward (along the positive $y$ axis) with focus at the origin, and with polar axis being the $x$ axis is
$ r_1(t) = \dfrac{ 2 p }{ 1 - \sin t } $
where $p$ is ...
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Proving that the point is fixed
Let $(O)$ be a circle.Each point $P$(other than $O$) corresponds to a line $p$, called the polar of $P$ with respect to $(O)$, which is perpendicular to $OP$. Conversely, every line $p$ (not passing ...
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Proving that the point is fixed
This was not as easy as I thought.
Draw the second tangent (other than $CP$) from $C$ to the circle with centre $O$ and call it $CJ$. Define $E'$ as the intersection of $AB$ and $PJ$.
Claim: $E = E'$....
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Find Matrix Describing The 2D Ellipse Made by Intersecting a 3D Ellipsoid w/ a Plane
Since the observer is far away, we can assume the observer is at infinity.
Therefore what we're looking for is the orthogonal projection of the ellipsoid onto the plane $n^T p = d$ (we can take $d = ...
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Prove that the segment $BE$ intersects the common point of two circles
Let $C \neq X = C_1 \cap C_2$. Note that
$$\begin{align} \angle BXC &= 180^\circ -\angle BAC
\\ &= 180^\circ -\angle ACB \\
&= 180^\circ - \angle ADB \\
&= \angle BDE
\end{align}$$
...
- 5,025
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Prove that the segment $BE$ intersects the common point of two circles
Let's mark the point of intersection of two circles as $P$. We know that $DC = DP$, as radiuses of $C_{2}$. Now let $E'$ be the point of intersection of $BP$ and $AD$. We want to prove that $E' = E$. $...
- 105
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Proving that the sum of two angles equals to 180°
Let $E$ be the intersection point other than $D$ of the circumcircles of $\triangle DAB$ and $\triangle DMK$. Then $E$ is the spiral similarity center which sends segment $KB$ to $MA$, and since $P$ ...
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How many intersections are there for two confocal parabolas with their axis perpendicular to each other?
There's a simple geometric argument for why it's not 4, but at most 2 points. While intuitively, I can see how it must be exactly 2 and not less, I don't have a clear geometric argument for that yet.
...
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Locus of centers of conics through four points by geometry
OP claims a solution has already been obtained by using a coordinate system, but it is not presented. For clarity, I will derive this here, and show that the midpoints are always on the conic. For ...
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vote
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Verify the formula for the the 3-simplex
We can argue inductively. If $S = \{\mathbf x \in \mathbb R^4 : \mathbf x \ge 0 \text{ and } x_1+x_2+x_3+x_4=1\}$ is the set we're trying to understand, then it has four faces given by the ...
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