0
votes
0answers
11 views

Constructing triangle using side length-median relationship

$m^2_a=\frac{2b^2+2c^2−a^2}4$ $m^2_b=\frac{2c^2+2a^2−b^2}4$ $m^2_c=\frac{2a^2+2b^2−c^2}4$ Solving for a,b,c in terms of $m^2_a,m^2_b,m^2_c$ gives: $a^2=\frac{8m^2_b+8m^2_c−4m^2_a}9$ ...
1
vote
1answer
23 views

Relationship between the altitude of an isosceles triangle and segments drawn to the lateral side from a point on the base.

Question :In an isosceles triangle, the sum of the distances from each point of the base to the lateral sides is constant. I've tried a couple of things, but it seems like this statement is not true. ...
1
vote
2answers
27 views

Relation of length of a projection of a point to a line

In the given figure, can it be said that $x \leq a + b - d$?
0
votes
1answer
15 views

Determine sides of obtuse triangle

I really cannot figure this question out. Can anyone give me a hint please!? Find an integer $a$, for which $a$, $a+1$ and $a+2$ are the lengths of the sides of an obtuse triangle.
0
votes
0answers
17 views

Cut RAIT pieces from a RAIT cake

We baked a cake shaped like a RAIT (= right-angled isosceles triangle). We sprayed a large number ($N$) of raisins arbitrarily over the cake. Now we want to give each of our two children a piece that ...
0
votes
1answer
54 views

Disprove the possibility of such a triangle.

The image is not that good, but, consider the following figure to be true without actually constructing it,how can one person find a $fault$ in it. The blue colour represents perpendicular, The ...
0
votes
1answer
36 views

How to find coordinates of 3rd vertex of a right angles triangle when everything else is known?

I want to locate precisely the 3rd coordinate of a right angled triangle. I have: the length of three sides The three angles The other two coordinates of the triangle The triangle can lie in any ...
1
vote
0answers
16 views

Self-tiling tile-sets of order 2

A rep-tile is a geometric shape that can be partitioned to smaller copies of itself. The order of a rep-tile is the number of small copies. E.g., a square is a rep-tile of order 4. The smallest ...
0
votes
1answer
16 views

Calculation of third vertices of a triangle given a vector that should be perpendicular to the triangle plane

I have an isosceles triangle (2 sides same length) with vertices O, A and B. OA and OB are the same length. Vertices O and A are known, with O at origin (0,0,0). A known vector V, should be ...
0
votes
1answer
28 views

Find the area of the triangle [closed]

Let $P=(0,1,0),\ \ Q=(-1,1,2) \ $ and $ \ R=(2,1,-1)$. The answer is $1.5$. How is this solution found?
0
votes
0answers
30 views

Infinite Sum of Areas of Isosceles Triangles Help

Do the areas of the isosceles triangles form a geometric series? I can't figure it out! Triangle $\Delta ABC$ is formed such that $\overline{AC}$ is on the x-axis, $\overline{AB}$ is on the line ...
0
votes
0answers
22 views

Isosceles triangle's angles calculation

How can I calculate angles in an isosceles triangle? I have an isosceles triangle ($a=b=10$). Is there any general formula that can be applied to all isosceles triangle's angles without having to ...
0
votes
1answer
27 views

If a triangle has side lenghts $a,b,c$ where $c$ is the largest prove that its obtuse if $c^2>a^2+b^2$ and acute if $c^2<a^2+b^2$.

I was thinking about this and I cant get to a formal proof. I have a sort of mental image where you draw $a$ and $b$ perpendicular and the $c$ is too small to connect the two endpoints. So the right ...
0
votes
1answer
22 views

The amount of unit squares being covered

$L$ and $i$ are integers, $L$ is the length of edge of outermost square and $i$ is the minimum length divided from $L$. And there are cells or unit squares consisting the whole block. There is a ...
5
votes
2answers
149 views

What is the flaw in this proof that all triangles are isosceles?

What is the flaw in this "proof" that all triangles are isosceles? From the linked page: One well-known illustration of the logical fallacies to which Euclid's methods are vulnerable (or at least ...
1
vote
2answers
50 views

A simple geometry question

Suppose $ABC$ is any triangle and $BE$ is any line from the vertex $B$ to a point $E$ lying inside the segment $AC$. Let $D$ be any point on $BE$. I would like to verify the following: regardless of ...
0
votes
0answers
20 views

Create dynamic cities of perspective angle x

I'm creating a tilemap... I found you can create unique building sizes with perspective with six tiles using parallel projection, whose angles are always 45 degrees... this allows you to connect to ...
3
votes
3answers
56 views

Right-Angled Isosceles Triangle covering puzzle

Consider a RAIT (right-angled isosceles triangle), from which we remove a RAIT smaller than half its area by a cut perpendicular to the hypotenuse, like this: How many RAITs are required to cover ...
16
votes
7answers
2k views

Can one deduce whether a given quantity is possible as the area of a triangle when supplied with the length of two of its sides?

Recently I have found a question like following: In triangle $ABC$, $AB=AC=2$. Which of the following could be the area of triangle $ABC$? Indicate all possible areas: [A] $0.5$ [B] $1.0$ ...
1
vote
2answers
40 views

Prove triangles formed by two midpoints and an altitude are congruent

Triangle ABC has altitude BH. M is the midpoint of AB, and N is the midpoint of CB. Prove triangle MBN is congruent to triangle MHN. Can we say that MN bisects BH? If so, why? If MN bisects BH (at ...
1
vote
1answer
44 views

Prove triangle made from two altitudes and midpoint is isosceles

In triangle ABC, AH and BK are altitudes. M is the midpoint of AB. Prove that triangle MHK is isosceles. All I can see is that the angles formed where the altitudes intersect are equal, and since ...
0
votes
2answers
68 views

In any triangle ABC, the expression (a + b + c) (a + b - c) (b + c - a) (c + a - b)$ is equal to

In any triangle ABC, give an equivalence to the expression $$(a + b + c) (a + b - c) (b + c - a) (c + a - b)$$ Can somebody help me? Note that ...
7
votes
4answers
187 views

Construction of a triangle

I need to construct a triangle with given information: $c = 6$, $h = 4$ and $\alpha - \beta = 30º$. I'll put approximate result for any clarification.
1
vote
1answer
39 views

Calculate PQ if AC = 20

I need to calculate PQ knowing that AC = 20. This is what I got so far: If I call the point between P and A, "M" and If I call the angle: $$\measuredangle{QPB} = y$$ Then: ...
0
votes
2answers
51 views

Find the value of $a$.

please help I'm lost on what numbers to add or what formula to use
2
votes
1answer
32 views

Do the medians (or other cevians) form all the triangles?

I want to know whether set of medians of all triangles, or some other class of cevians, can form the set of all the triangles? For example, in the case of altitudes, $(4,7,10)$ is an counterexample. ...
0
votes
1answer
33 views

Geometry, Mensuration

If the diagonal BC passes through center of the circle, then the area of the shaded region in the given figure is \begin{align*} a)\quad &\dfrac{a^2}{2(3-\pi)}\\ b) \quad ...
1
vote
1answer
23 views

$S$, $I$, $O$ are circumcenter, incenter and orthocenter then $SO\ge IO \sqrt2$

Let $S$, $I$ and $O$ be the circumcenter, incenter and orthocenter of $\triangle ABC$ then prove that $SO\ge IO \sqrt2$, or equivalently $SO^2\ge 2IO^2$. I was able to derive an expression for $SO^2$ ...
2
votes
1answer
40 views

Sum of inradius of constructed triangle

Let $ABC$ be a triangle with inradius $r$ and circumradius $R$. Let $A′B′C′$ be the triangle for which $A′B′$ is the perpendicular to $OC$ through $C$ and so on. Let $r_1$ be the inradius of $A'BC$, ...
0
votes
2answers
49 views

Maximum Area of a Triangle when 1 Side, Perimeter Known

This is an example of a "quantitative comparison" question the GRE would test. Suppose the following information is known: one side of a triangle has length 12 the perimeter of the triangle is 40 ...
2
votes
2answers
44 views

How to find the area of an isosceles triangle without using trigonometry?

I have an isosceles triangle with equal sides $10$ unit, angle between them is $30^\circ$. I need to be confirmed that the area of this triangle can be found in any method other than using any kind ...
4
votes
4answers
82 views

Construction of an equilateral triangle from two equilateral triangles with a shared vertex

Problem Given that $\triangle ABC$ and $\triangle CDE$ are both equilateral triangles. Connect $AE$, $BE$ to get segments, take the midpoint of $BE$ as $O$, connect $AO$ and extend $AO$ to $F$ where ...
2
votes
3answers
53 views

For a triangle $ABC$, $a^2+b^2+c^2=8R^2$ then it is a right triangle?

$ABC$ is a triangle, $a^2+b^2+c^2=8R^2$ then how do we prove it is a right triangle?
0
votes
1answer
48 views

Triangle question, proving isoceles given trigometric conditions

$ABC$ is a triangle satisfying the following condition: $$\frac{\sin B}{\sin A}=\frac{\tan B+\cot C}{\tan A+\cot C}$$ How do I prove that $ABC$ is isoceles? I really have no idea.
0
votes
1answer
33 views

How prove that $|QA| < |QC|$ in triangle?

$ABC$ is a triangle with a right angle at $A$, and $|AB|$ > $|AC|$. The point $D$ is defined so that $BCD$ is equlateral and $AD$ intersects $BC$ at $P$. The point $Q$ is defined so that $QDP$ is ...
0
votes
2answers
33 views

How do I find a missing angle using a reciprocal trigonometric function?

I just attempted this as best as I could, but I'm not sure if I'm correct. Here's the work: $$\cot x =\frac{1}{2}$$ $$\frac{1}{\tan{x}} = \frac{1}{\frac{1}{2}}$$ $$\frac{1}{\tan^{-1}\cdot\tan x} = ...
2
votes
1answer
29 views

If $\frac1{HB}-\frac1{HA}=\cot C \cdot (\frac1{BC}-\frac1{AC})$, where $H$ is the orthocenter, then $ABC$ is isoceles?

If given that for a triangle $ABC$, with orthocenter $H$:$$\frac1{HB}-\frac1{HA}=\cot C \cdot (\frac1{BC}-\frac1{AC})$$ Then prove or disprove that $BC=AC$. How should I proceed with this?
4
votes
1answer
78 views

How show that $ABC$ is equilateral?

Let $D$, $E$ and $F$ be three points on sides $BC$,$AC$ and $AB$ of triangle $ABC$ such that lines $AD$, $BE$ and $CF$ concur at point $M$. If three trianles $MDB$, $MCE$ and $MAF$ have equal areas ...
1
vote
1answer
44 views

How prove that $AD>BE$ in triangle?

Let $D$ be a point on the side $BC$ of a triangle $ABC$ such that $AD>BC$ . The point $E$ on $CA$ is defined by the equation $\frac{AE}{EC}=\frac{BD}{AD-BC}$ .How prove that $AD>BE$?
3
votes
2answers
56 views

Geometrical proof for $PA+PB+PC\le3R$, where $P$ is the orthocenter and $R$ is the circumradius

$ABC$ is an acute angled triangle, where $P$ is the orthocenter, and $R$ is the circumradius. I want to show that $PA+PB+PC\le 3R$ geometrically, that is without using trigonometry. I have a trig ...
1
vote
1answer
44 views

Splitting a triangle to make two equal halves, find the length of the new line

Could someone please explain to me how I would find this out? I have a triangle and I need to find the length of the line that would split it down the middle so that the areas were even. A = 105 ...
2
votes
2answers
252 views

Area of Triangle when 2 Sides and No Angle Known

It is quite possible this question has no answer -- that is, the area cannot be determined from the information given. It's a question I've created myself as I study for the GRE. No trigonometry is ...
1
vote
2answers
41 views

Find side BC of a triangle given AB, AC, and a relation between $\angle A$ and $\angle B$

A question from my class: In triangle $ABC$, $3\angle A+2\angle B=180$ and $AB=10, AC=4$. So question is, what all can we comment on side $BC$. Can we find its exact length? I have a crude ...
2
votes
0answers
54 views

How to prove that three points are collinear [closed]

If H is the point within triangle ABC prove that the external bisector of the angles of AHB, BHC, CHA meet AB, BC, CA respectively at three collinear points. I don't have any idea how to solve this ...
-1
votes
1answer
97 views

How to get the third point coordinates in isosceles triangle?

Isosceles triangle $ABC$ $AB = AC = d_1$ $BC = d_2$ $A = (x_1, y_1)$ $B = (x_2, y_2)$ $C = (x_3, y_3)$ $\angle BAC = \phi$ $\angle ABC =\angle ACB = \theta$ I want an equation for $x_3$ and $y_3$ ...
0
votes
1answer
53 views

Number of triangles formed by all chords between $n$ points on a circle

We have $n$ point on circumference of a circle. We draw all chords between this points. No three chords are concurrent. How many triangles exist that their apexes could be on circumference of ...
1
vote
1answer
52 views

problem about length of perpendicular chords

Question $AB$ is chord of circle $O$,points $D$ and $E$ are chosen on $AB$ in a way that $AD=BE$.prove two chords that are perpendicular to $AB$ and pass $D$ and $E$ points are equal.(prove $LK=MN$) ...
0
votes
1answer
22 views

Triangle Theorem relating the shortest and longest distance from any arbitrary point inside

I recall somewhere there was a relationship such that given a triangle T and a point A: if A is inside of T, then the sum of the longest distance from A to any point on a side of T, plus the shortest ...
5
votes
1answer
51 views

Circle with perpendicular chords

A blue circle is divided into $100$ arcs by $100$ red points such that the lengths of the arcs are the positive integers from $1$ to $100$ in an arbitrary order. Prove that there exists two ...
1
vote
1answer
92 views

Minimum Value of $x_1+x_2+x_3$

For an Acute Triangle $\Delta ABC$ $$\begin{align}x_n=2^{n-3}\left(\cos^nA+\cos^nB+\cos^nC\right)+\cos A\,\cos B\,\cos C\end{align}$$ Then find the least value of $$x_1+x_2+x_3$$ My Approach: I have ...