0
votes
1answer
19 views

Having trouble with showing this CANNOT be a theorem in incidence geometry.

Consider the following statement: If l and m are any two distinct lines, then there exists a point P that does not lie on either l or m. (a) Show that this cannot be a theorem in incidence geometry. ...
4
votes
4answers
296 views

How to show that these two lines are perpendicular?

Let $AEE'$ be an isoceles triangle with $\angle EAE'=90^\circ$ such that $AE=AE'$ and such that $A$, $E$ and $E'$ lie on the circle $c_1$. Let $ADD'$ be an isoceles triangle with $\angle ...
0
votes
1answer
58 views

geometry circle proof

Use a common notion to prove the following result: If P and Q are any points on a circle with center O and radius OA then OP is congruent to OQ. Since O is the center and P & Q are any where on ...
0
votes
0answers
64 views

Similar Triangles Proof - How to tackle proofs?

Well, I know it is repetitive.I have read the proof from different textbooks.But sometimes I feel doubtful about it all.Every time I try to prove it for myself, I fail at some points.I'm asking those ...
1
vote
1answer
32 views

Prove that line segments are parallel.

Prove using slope of lines that line segment joining the midpoint of $\overline { AB}$ and $\overline{AC}$ in $\Delta ABC$ is parallel to $\overline {BC.}$ Need to prove using slope of lines means I ...
0
votes
0answers
20 views

Every polyhedron $P \ne \mathbb{K}^n$ equals an intersection of finitely many half spaces.

Currently, I am reading some lecture notes on linear optimisation. I cannot see why the following (seemingly trivial) proposition holds. (How could I understand/proove it?) Every polyhedron $P \ne ...
1
vote
0answers
42 views

From 2 to 3 dimensions: integrating a force along a contour/surface.

I am studying the following problem: Consider a closed contour $\mathcal{C}$ in $\mathbb{R}^2$ defined by $r(\theta)$ where $\theta\in[0,2\pi)$ and $r(0)=r(2\pi)$ (let the center to be zero for ...
3
votes
1answer
108 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
39 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
40 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
33 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
177 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
49 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
136 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
62 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
40 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
75 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
37 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
74 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
74 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
160 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
80 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
106 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
86 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
41 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. ...
5
votes
3answers
453 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
118 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
105 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
72 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
172 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
92 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
71 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
197 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
62 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
66 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
149 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
24 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
93 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
209 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
1k 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
51 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
60 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
164 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
60 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
85 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
2k 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
156 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
231 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
134 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.