3
votes
1answer
74 views

Prove that there is only one way to make a square using all six tangram pieces

I am pretty sure there is only one way to make a square from the six tangram pieces: How can I prove this is the only way respecting all symmetries?
1
vote
1answer
24 views

Proof of isometries and inverses on the plane

I am taking a course on Intuitive Geometry. I am quite new to intuitive proofs however feel I've done pretty well thus far. Here is my theorem: Prove: That every isometry has an inverse. $Proof.$ ...
0
votes
0answers
27 views

Issue with the geometric proof of lim_{x -> 0} sinx/x = 1

When proving $\displaystyle\lim_{\theta \to 0} \frac{\sin\theta}{\theta} =1$, I have been taught to use a sector with radius 1. How rigorous is this proof if we have not considered a radius of any ...
2
votes
1answer
32 views

$\frac{|| \overline{AM}||}{|| \overline{AB}||}=\frac{|| \overline{AN}||}{|| \overline{AC}||}=\frac{|| \overline{MN}||}{|| \overline{BC}||}$

$\Delta ABC$ is a triangle, $M$ is a point in the segment $\overrightarrow{AB}$ and $N$ is a point in the segment $\overrightarrow{AC}$, such that $\overrightarrow{MN}$ is parallel to ...
4
votes
1answer
155 views

How to prove that equilateral triangle formed by cube's corners cannot be fully inserted to this cube

I would like to prove that equilateral triangle prescribed by cube's corners and sides equal to $b = a \sqrt{2}$ cannot be inserted into the interior of a cube of side $a$. This triangle is presented ...
2
votes
1answer
41 views

Square Line Picking

The probability density function of the distance between two points chosen randomly on the unit square is given by: $ P(\ell) = \begin{cases} 2\ell\left(\ell^2 - 4\ell + \pi\right) & 0 \leq \ell ...
7
votes
2answers
123 views

Prove That the Second Moment is Minimized with a Circle Packing

Graham and Sloane studied the problem of minimzing the second moment of disks on the plane, i.e. minimize $$ U = \frac{1}{d^2} \sum_{i=1}^{n} || \mathbf{p}_i - \bar{\mathbf{p}} ||^2 $$ s.t. ...
0
votes
2answers
55 views

Proving the inscribed angle theorem

I need to prove that a circle's inscribed angle is 1/2 of the arc it intercepts. I am given that one of the chords making up the angle is the diameter. I have an entire project to do based off of this ...
0
votes
0answers
35 views

2 column proof of The Tangent-Chord Angle Corollary

I need to prove 12.23 on this section (http://i.imgur.com/M5iev9K.png) I can use any of the theorems or corollaries before 12.23 but not the ones after it. This is a list ...
0
votes
2answers
70 views

How to show that a regular pentagon can't have all coordinates rational

This is pretty straightforward if we're allowed to use trigonometry, so I guess my question is Are there any nice (trigonometry-less) proofs of the fact that a regular pentagon in the plane must ...
1
vote
2answers
33 views

How to prove that certain points relating to a trapezoid are collinear?

Can you help me to prove that in any trapezoid, which is not a parallelogram, the following points are collinear? The midpoints of its bases. The point of intersection of diagonals. The point of ...
3
votes
2answers
70 views

Is a quadrilateral with one pair of opposite angles congruent and the other pair noncongruent necessarily a kite?

If convex quadrilateral ABCD has congruent angles A and C and the other pair of angles B and D are not congruent, is ABCD necessarily a kite (two pairs of consecutive congruent sides but opposite ...
1
vote
2answers
70 views

$ABCD$ right angle trapezoid

Let $ABCD$ be a right angle trapezoid, with angle $\hat{A} =90$ degrees and $\overline{AB}$ is parallel to $\overline{CD}$. Let $O$ be the intersection point of the diagonals $\overline{AC}$ and ...
2
votes
2answers
101 views

Let ABCD be a trapezoid, such that AB is parallel to CD.

Let $ABCD$ be a trapezoid, such that $AB$ is parallel to $CD$. Through $O$, the intersection point of the diagonals $AC$ and $BD$ consider a parallel line to the bases. This line meets $AD$ at $M$ and ...
0
votes
1answer
70 views

Prove that every half-plane is a nonempty set.

So, I have no idea how to prove this. In my mind, I guess I'm thinking if you just had a line that made the half-planes and nothing else then why couldn't it be empty? If anyone can giveme an didea on ...
1
vote
2answers
104 views

Two parallelograms are equal in area.

I tried this question by constructing a line $PD$ therefore forming two triangles $ADP$ and QDP but couldn't establish the congruency relation between the triangles. My approach was that if I have ...
1
vote
2answers
72 views

In triangle $ABC$ prove that $AB = 2BC$

In solving this proof I am NOT permitted to use any numerically related givens (i.e., the sum of all angles in a triangle is 180 degrees or in a right triangle side Asquared + side Bsquared = side ...
0
votes
1answer
38 views

(Geometry) Proof type questions

Can someone please explain to me the given question and proof? otherwise I might just have to end up dropping my maths course because unfortunately I'm not understanding anything from my teacher. ...
4
votes
3answers
311 views

Proving the area of a square and the required axioms

I recently realized the area formula of all polygons, and most basic figures can be proven from the areas of square and rectangle. For example if we know the area of rectangle, we can the area formula ...
2
votes
1answer
100 views

A proof of an interesting Geometric Vector Theorem.

Suppose $O$ is the centre of the circumscribing circle of triangle $ABC$ and $H$ is its orthocentre. Prove that vector $OH$ is equal to the sum of the vectors $OA$, $OB$ and $OC$. An answer I ...
5
votes
1answer
91 views

Rational distance from an equilateral triangle

Is there a nice proof for the following fact? In a plane, there does not exist a square such that its vertices are at a rational distance from each vertex of some equilateral triangle. What if ...
2
votes
0answers
71 views

A follow-up to the regular hexagon question

This is a follow-up to the regular hexagon question. The problem statement was: Suppose we have a sphere and more than a half of its surface is red. Prove or disprove that we can place all ...
1
vote
3answers
145 views

A Proof with Intersecting Lines

My geometry teacher has given me a question to try to solve which is: Prove that there exists lines a and b, such that a is not equal to b and a intersects b. I am not sure how to prove this or ...
1
vote
0answers
86 views

Solving construction problems?

I recently encountered 'construction problems' in geometry. These were quite new to me and I didn't know the requirements they expected and prerequisites to solve them. I'll explain with an example. ...
1
vote
2answers
69 views

Proof using properties of an isosceles or right-angle triangle

Given a triangle $ABC$ with sides $AB=BC$ and angle$\angle B=100^\circ $, prove that $$a^3 + b^3 = 3a^2b$$ where $a=AB=BC$ and $b=AC$, I have tried to use simultaneously the sine and cosine rules as ...
1
vote
2answers
176 views

Proof of geometric congruence using linear algebra

We may assume some set $T$ of all triangles within the same plane. Let $R$ be defined on $T$ where $a\ R\ b$ if the triangles $a, b$ are congruent. We may assume congruence to be defined as follows ...
2
votes
1answer
55 views

Does the fact that a tiling is tile-uniform always guarantee that it is also vertex-uniform?

It seems to me that if a tiling is tile-uniform, then it must be vertex-uniform as well. But is this the case? How would one go about devising a proof? By 'tile-uniform', I mean a tiling whose ...
2
votes
2answers
63 views

Should I be able to prove Law of Cosines, Half Angle formula, etc?

This is more of a general question then the title suggests, but the laws in the title are what I'm currently studying. I can read the proofs of both and understand them after a while, but I could ...
3
votes
2answers
125 views

Tallest bubble tower induction proof

A hemispherical bubble is placed on a spherical bubble of radius $1$. A smaller hemispherical bubble is then placed on the first one. This process is continued until $n$ chambers, including the ...
0
votes
1answer
23 views

Proof with Segments

I have an assignment (from a tutor) that tells me: Give an informal proof for "If A-B-C and point P is on segment AC, then P is on segment AB or P is on segment BC" To start my proof, I have assumed ...
0
votes
1answer
81 views

Proving that two crossbars of a bisector intersect the midpoint of one of the crossbars

This is sort of an involved question. I've proved parts of it already but now I'm stuck. Here is the question: Let ∆ABC be such that AB is not congruent to AC. Let D be the point of intersection of ...
0
votes
2answers
157 views

Transitivity of parallel lines

I cam across a question (in my textbook) about proofs with parallel lines. The question is: Prove that the property that || is transitive implies that for any point P and line l, there is at the most ...
0
votes
1answer
741 views

Proof for Symmetry property of Congruent Segments

I am starting to learn geometrical proofs, and I have come across the Symmetry property of segment congruence (if $AB$ is congruent to $CD$, then $CD$ is congruent to $AB$). One of the exercises in ...
0
votes
1answer
50 views

Is simple straight-edge and compass construction a substantial proof?

I'm working on a problem that asks to prove that a point $D$ is outside of a $\triangle ABC$, on the circle through the triangle, given that sides $AB$ and $AC$ are not congruent, and that $D$ is the ...
1
vote
1answer
59 views

Geometry Proof Triangles

Show that if two of the corresponding angles of two triangles are equal then so is the third. Is there a formal way to prove this? I wanted to just say in one sentence that if two angles are the ...
1
vote
2answers
138 views

How to prove point A belongs to line t?

I'm stuck at trying to prove that any point $A$ will belong to line $t$ if and only if segments $AB=AC$, where $B$ and $C$ are symmetrical points to the line $t$ and $M$ is the midpoint of segment ...
0
votes
1answer
59 views

proof, for a differentiable function from $\mathbb{R}^n$ to $\mathbb{R}$, $\int_{Cpq}{\nabla f\cdot\mathrm{d}\boldsymbol{r}} = f(q)-f(p)$

For a differentiable function: $f:\mathbb{R}^n\rightarrow\mathbb{R}$ prove that: $$\int_{C_{\boldsymbol{pq}}}{\nabla f}\cdot\mathrm{d}\boldsymbol{r}=f(\boldsymbol{q})-f(\boldsymbol{p})$$ where $C$ ...
2
votes
2answers
80 views

Can a cyclic quadrilateral be inscribed in a parabola?

It is quite obvious that a quadrilateral can be inscribed in a parabola. However, can somebody provide a nice (meaning: intuitive) proof that a cyclic quadrilateral can be inscribed in it? Further, ...
1
vote
3answers
1k views

Vectors: Prove that if median in a triangle is perpendicular to the corresponding base, then the triangle is isosceles.

Prove using vector methods that if a median in a triangle is perpendicular to the corresponding base, then the triangle is isosceles. I know the basic idea. The dot product of the median vector and ...
4
votes
2answers
154 views

A proof in circles.

I need help proving this problem: $AB$ is a diameter of a circle. $CD$ is a chord parallel to $AB$ and $2CD = AB$. The tangent at B meets the line $AC$ produced at $E$. Prove that $ AE = 2AB $. ...
0
votes
1answer
211 views

$n$ Lines in the Plane

How am I to "[u]se induction to show that $n$ straight lines in the plane divide the plane into $\frac{n^2+n+2}{2}$ regions"? It is assumed here that no two lines are parallel and that no three lines ...
2
votes
1answer
124 views

Pythagorean theorem proof by dissection

I have this proof of the Pythagorean theorem, but in the last two lines of the fourth paragraph I can't seem to find geometrically how the congruency between $1$, $2$, $3$, $4$ and $1'$, $2'$, $3'$, ...
1
vote
1answer
52 views

Prove that a polygon with nonnegative area is determined by at least three points.

How do you prove this statement in geometry? A polygon with nonnegative area can't be formed with fewer than 3 points.
1
vote
0answers
103 views

Proving that the circumcenter is the centroid

Given a triangle and its centroid, we know that the 3 line segments between the centroid and each of the vertices of the triangle divide the triangle into three smaller triangles. Prove that the ...
5
votes
3answers
234 views

Pythagorean theorem and its cause

I'm in high school, and one of my problems with geometry is the Pythagorean theorem. I'm very curious, and everything I learn, I ask "but why?". I've reached a point where I understand what the ...
0
votes
1answer
82 views

Proof: Two circles have a most 2 intersections

I already prooved the statement here in general, but know I tried to proove it in an other way: I put $M_1$ on $(0/0)$ and the x-axis through $M_1$ and $M_2$. That simplifies the equatons for ...
0
votes
2answers
92 views

Proof pythagoras theorem with dot product + distance

I want to proof $d(A.B)^2=d(A,C)^2+d(B,C)^2$ for with $(\vec a-\vec c) \bullet (\vec b - \vec c)=0$. I applied the definitions of distance and got $d(A,B)^2=d(A,C)^2+d(B,C)^2 \Leftrightarrow ...
0
votes
2answers
77 views

Are the lengths from this recursive construction a geometric sequence?

In his 1999 review of Edward Tufte's Visual Explanations in the Notices of the AMS (third page), Bill Casselman gives a very pretty proof of the irrationality of the golden mean. More precisely, ...
6
votes
0answers
374 views

Proof Strategy for a Dynamical System of Points on the Plane

I have a rather simple-looking system which exhibits a particular behaviour in simulation, and I would now like to attempt to prove this formally. The problem is, I don't really know where to start, ...
0
votes
1answer
43 views

Euclidean space problem

In three-dimensional space, is it true that if you take line $a$ of a plane and line $b$ of the plane perpendicular to the first one, then the angle between line $a$ and $b$ (at which they intersect) ...