For questions about triangles

learn more… | top users | synonyms

1
vote
2answers
37 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
12 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 ...
8
votes
5answers
93 views

Are all triangles where “$a^2 = b^2+ c^2$”, right-angled?

For a right angle triangle, you can say that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Does the converse hold, ie. can you also say that, if the square ...
1
vote
2answers
58 views

Proving that an equilateral triangle in the plane cannot have vertices on integer lattice points

Thanks for the help! I've written a more detailed proof. The hints were great.
-1
votes
3answers
29 views

Triangle Inequality on complex numbers

Problem Let $z= x + iy$, then prove that: $$|x| + |y| \le 2 ^{1/2} |z|$$ Progress I've tried to write $|z|$ as $(x^2 + y^2)^{1/2}$, and to make some algebra after this, but I'm really new at ...
-5
votes
0answers
20 views

<html5> draw circle by arc and triangle [closed]

I want if click the retacgle, draw a triangle around the circle. source code like this... but, triangle was not good each of positions.. How can i draw a triangle around the circle like attached ...
1
vote
0answers
28 views

How find a triangle ABC minimizing $\frac{\sqrt{1 + 2\cos^2 A}}{\sin B} + \frac{\sqrt{1 + 2\cos^2 B}}{\sin C} + \frac{\sqrt{1 + 2\cos^2 C}}{\sin A}$?

How find in triangle $ABC$ the minimum value of : $$\frac{\sqrt{1 + 2\cos^2 A}}{\sin B} + \frac{\sqrt{1 + 2\cos^2 B}}{\sin C} + \frac{\sqrt{1 + 2\cos^2 C}}{\sin A}\text{ ?}$$
3
votes
3answers
48 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 ...
-2
votes
0answers
40 views

Straight lines - pair of lines [closed]

Question: Let PQR be a right angled triangle with right angle at P(2, 1). If the equation of the line QR is $2x+y=3$, then the equation representing the pair of lines PQ and PR is: a) $3x^2 - 3y^2 + ...
16
votes
6answers
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
57 views

Trigonometry and triangle proof

Question: Prove that in an acute angle triangle ABC: $$\tan A\tan B +\tan A \tan C + \tan B \tan C \geq 9$$ I have no idea where to even begin this question. Please help me!
1
vote
2answers
34 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
42 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
174 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: ...
1
vote
4answers
95 views

Ratio of Areas of Similar Triangles

First step, I can't find the height. How do you find the height?
0
votes
2answers
49 views

Find the value of $a$.

please help I'm lost on what numbers to add or what formula to use
1
vote
1answer
26 views

Fourier transform of a triangular pulse

So I've been practicing some fourier transform questions and stumbled on this one; To start off, i defined the fourier transform for this function by taking integral from -tau to 0 and 0 to tau as ...
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
21 views

Translate line vertically and calculate intersection on circle

Let's say I have a line extending from the center of the circle at a 45° angle. If I were to translate that line up 212.132 units, how would I calculate the intersection between the translated ...
0
votes
1answer
29 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 $SO^2$ ...
1
vote
0answers
13 views

Complete Triangle Given 3 Parallel Planes and 2 Points

I have a problem where a point B connects to a point C at a known angle and distance. Both point B and C are on two separate parallel axis, GH and JK respectively. I need to find a third point, A, on ...
0
votes
2answers
35 views

isosceles and tight triangle

Hi, I was wondering if there is a way to find x with only knowing the length of isosceles triangle and no other piece of information.
2
votes
1answer
36 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
40 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
40 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 ...
0
votes
0answers
34 views

Translate vertical movement into radial movement?

I've tried all sorts of things, but I'm no mathematician and I've conceded defeat. So I come here for help. I don't know if I really worded the question correctly since I don't even know what I should ...
4
votes
4answers
81 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
45 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
30 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
24 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
77 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
43 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
52 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
34 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
235 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
37 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 ...
1
vote
2answers
52 views

Find the angle between the sides 4 and 7 in a right triangle

I need to solve the $B$ corner What I've tried: $$\operatorname{sin} B=\frac47$$ $$B=\operatorname{arcsin}\frac47$$ $$B=34.85$$ But that's not the right answer, can anyone help me find what I did ...
2
votes
0answers
52 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
2answers
52 views

Proving the following inequality in a triangle

In a triangle the straight lines $AD$, $BE$, $CF$ are drawn through a point $P$ to meet $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively: Prove that $$\frac{PD}{AD} + \frac{PE}{BE}+\frac{PF}{CF}=1$$ ...
1
vote
1answer
18 views

Incentre of the triangle proving

A straight line is drawn through the incentre I of the triangle ABC perpendicular to AI meeting AB, AC in D and E respectively. Prove that BD.CE=ID^2
0
votes
4answers
47 views

How to find an angle (in degrees) in a right triangle, given its sides?

I need to find out a degree of an angle. Pretty simple, or so I thought. I remember doing a crap-ton of these in high-school, sadly the details did not remain. Anyway, let's take a look at this ...
1
vote
3answers
31 views

Basic question about angles

Why is the answer a)? Why can't it be d)? Why are the choices listed in this format, i.e., $(x \pm \theta^{\circ})$, and why is angle C $(x+30^{\circ})$ and not just $30^{\circ}$? Thanks.
2
votes
2answers
71 views

Simple proof of existence of hyperbolic triangles

I've studied the hyperbolic plane by analytically building up the hyperboloid model, the Klein—Beltrami disc, the Poincaré disc, and the half-plane model from scratch. Now I'd like to prove that, ...
-1
votes
1answer
71 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
47 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 ...