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Questions tagged [triangle]

For questions about properties and applications of triangles.

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1answer
26 views

Triangle vector problem

I consider how to draw this because I have no idea with the Origin, is the 3.1 part we can solve with : triangle QAB: triangle QBC so, AQ : QC = 2:3 ? But I can't solve the 3.2 part, can ...
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3answers
41 views

Triangle Inequality help

Wondering where my logic is going wrong in this assignment: Show that $||x|-|y|| \leq |x-y|$ Using the fact $||x|-|y||, |x-y| \geq 0$ It follows $(|x|-|y|)^2 \leq (x-y)^2$ Using the fact $|x|^2 = ...
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1answer
25 views

A problem about minimizing the area of a triangle within the coordinate plane.

The problem goes as follows: Given a line $l:y=ax$, where $a\in \Bbb{R}$ and $a>0$, and a point $P(b,c)$ that does not lie on $l$. Let $M$ be a point on the x-axis, and $Q$ the intersection of line ...
2
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1answer
27 views

Building a triangle from random segments [duplicate]

The problem Let's consider the interval $[0, 1]$. We break this interval randomly in two parts. Then we choose the bigger part and again we break it randomly into two parts. At the end we have three ...
2
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1answer
34 views

Calculate the ratio of the sides of a given triangle given the ratio of areas.

Given a triangle $\triangle ABC$, points $M$, $N$, $P$ are drawn on the sides of the triangle in a way that $\frac{|AM|}{|MB|} = \frac{|BN|}{|NC|}= \frac{|PC|}{|PA|}=k$, where $k>0$. Calculate $...
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2answers
41 views

Further Question on “ What is the probability that the center of the circle is contained within the triangle? ”

Q: Consider the triangle formed by randomly distributing three points on a circle. What is the probability of the center of the circle be contained within the triangle? This question was raised by ...
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2answers
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About sum of right-angled triangles'area and area of circle.

the image shows right-angled triangles in semi-circle In Definite Integration, we know that area can be found by adding up the total area of each small divided parts. So, base on the Definite ...
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1answer
38 views

The range of values of $a$ such that…

Question The range of values of 'a' for which the common tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$ and the parabola $y^2=4x$ and their chord of contact can form an equilateral ...
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2answers
21 views

Proof regarding altitudes of a triangle and a midpoint of one of its sides

Let $\triangle ABC$ be a triangle with altitudes $\overline{AH}$ and $\overline{BK}$. Consider the axis of the segment $ \overline{HK}$. Let $M$ be the point of intersection between the axis and the ...
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1answer
34 views

In A trapezium ABCD, AB= 7cm, CD = 16cm, BE || AD is drawn. P,Q,R,S are mid point [on hold]

I got of area of BEC = 144 after this i calculated height and I got area of traingle ABD =126. In question options given are (a) 208 (b) 56 (c) 28 (d) 112 Is my answer right or wrong?
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2answers
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Calculate coordinates of third point in a triangle (2D) knowing 2 points coordinates [on hold]

Triangle I have 2 points v1 and v2. I have a length B. I can work out length A and therefore C if necessary. My aim is to find the coordinates of v3. I have tried a few different ideas but can't get ...
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1answer
22 views

Derivation of Area Formula in Coordinate Geometry

So I learned about this formula in coordinate geometry where if you know the coordinates of the three vertices in a triangle, you can calculate the area. Formula My question is how do you derive ...
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2answers
24 views

$AB_1$, $AB_2$, $AB_3$ are the altitude, angle bisector, median from vertex $A$ of $\triangle ABC$; arrange lengths $BB_i$ in ascending order

Consider an acute angled triangle $\triangle ABC$ such that $AB\lt AC$. If from $A$ altitude $AB_1$ is drawn, internal angle bisector $AB_2$ is drawn, and median $AB_3$ is drawn. ...
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1answer
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Cevians $AD$, $BE$, $CF$ are concurrent, as are cevians $DP$, $EQ$, $FR$; show that $AP$, $BQ$, $CR$ are concurrent

In $\triangle ABC$, $D$, $E$, and $F$ are points on $BC$, $CA$, and $AB$, respectively, such that $AD$, $BE$, and $CF$ are concurrent lines. Points $P$, $Q$, and $R$ respectively on $EF$, $FD$, and $...
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0answers
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How can I solve this word problem? [closed]

I was given this word problem to solve: The base of a ladder is 8 feet away from the edge of a building. The ladder is 17 feet long. How high up on the building does the ladder reach? I need ...
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4answers
32 views

How to prove $ \sqrt { \frac {s(c-a-b)}{2∆} } $ is equal to i?

Consider a right angles triangle. Let, c be it's hypotenuse and a,b it's other sides. Then prove, $ \sqrt { \frac {s(c-a-b)}{2∆} } $ is complex. Where ∆= area of the triangle and s is semi ...
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0answers
48 views

Find an angle without using trignometry

This question has already been asked before, so it is actually a duplicate of: How to make correct system of equations to solve for the angles in this triangle? But I was trying to solve this ...
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0answers
56 views

Parallel lines & triangles

Two lines $GE$ and $FD$ meet in $A$. They cut each other at $45$ degrees. Both $G$ and $F$ lie on the circumcircle of square $ALBK$ such that $E$ and $D$ lie on the $KB$ and $LB$ respectively without ...
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1answer
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Calculate bevel edge Icosphere

I have an Icosphere with 80 faces, 120 edges. Now i am looking to find out what the angle is of the bevel between all the faces. With the bevel i mean the following see the image below: So i am ...
3
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1answer
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Ceva's Theorem: Proving lines in a specifically constructed triangle intersect

Question: Let $o_1$, $o_2$, and $o_3$ be circles with disjoint interiors with centres $O_1$, $O_2$, and $O_3$, respectively. Among the lines tangent to both of the circles $o_2$ and $o_3$ there are ...
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0answers
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Resolving Angles B and C of an Isosceles Triangle whereas X can be any real number

I have a conundrum on my hands. The simple question is this: Given isosceles triangle ABC, whereas m∠B is equal to m∠C, and where m∠A is equal to $x+12$, what is the value of x? Within the context of ...
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0answers
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An intuitive explanation of calculating area of triangle above horizontal line on graph [closed]

What is an intuitive explanation of how, by using the horizontal line y = 2, we know that it stretches to hit x = 10. Do we need the function f(x) = 12 - x to know or could it be done only using the ...
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0answers
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Prove $AX$, $BY$, $CZ$ concur

$\Delta ABC$. Let $P,Q$ be two points lies in the interior of the triangle. Let $AP,BP,CP$ intersect $BC,CA,AB$ at $X_P,Y_P,Z_P$. Point $X_Q,Y_Q,Z_Q$ are defined similarly. Let $T_P$ be a point lies ...
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votes
3answers
59 views

geometry inequality

$M$ is a point in $\triangle ABC$. $AM$ intersect with $BC$ at $A_1$. $BM$ intersect with $AC$ at $B_1$. $CM$ intersect with $AB$ at $C_1$. Proof that: $$AA_1 \times BB_1 \times CC_1 \geq 27 (MA_1 \...
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2answers
33 views

these formulas only used for the right triangle?

Can I use the following relationships to calculate the sides of the below triangle (in the picture)? $tan5=\frac{AB}{6}$ $cos5=\frac{6}{AC}$ Can we use these formulas only for the right triangle? ...
0
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1answer
22 views

what is wrong with following right triangle's hypotenuse's square reasoning [closed]

In the image given in the link, the square of the hypotenuse is 2ab. But it can't be true. I can't be true. I can't seem to find the reason. any help would be appreciated.
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0answers
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+50

Did I apply the Ceva's theorem correctly to this problem?

I need to confirm the following solution. I'm making a mistake somewhere. But I can't find the error. I apply the trigonometric form of the Ceva's theorem: $$\frac{\sin \angle 3}{\sin \angle 4}× \...
3
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1answer
51 views

Show that three point $G,H,G_1$ are collinear.

Triangle $ABC$ has centroid $ G$ and orthcenter $H$. Line (through $A$) is perpendicular to $GA$, line (through $B$) is perpendicular to $GB$, line (through $C$) is perpendicular to $GC$ cut at ...
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1answer
21 views

Can you compute line segment that crosses a triangle?

I have a triangle with corner vertices (x1, y1, z1), (x2, y2, z2), (x3, y3, z3). I also have ...
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1answer
18 views

Bidirectionally of the “Tangent Criterion”

I've recently been reviewing some basic geometry concepts when I saw this one in Evan Chen's fantastic "Euclidean Geometry in Mathematical Olympiads" (EGMO). Proving $(i)\Rightarrow (iii)$ is quite ...
2
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1answer
29 views

What is the length of the hypotenuse?

We have $n$ isosceles-right-angled triangles. The hypotenuse of the $n^{\textrm{th}}$ triangle is the base of the $(n+1)^{\textrm{th}}$ triangle. For the first triangle, $T_{1}$, the length of the ...
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3answers
553 views

Area of a triangle inside an ellipse

$F_1$, $F_2$ are are foci of the ellipse $\dfrac{x^2}{9}+\dfrac{y^2}{4}=1$. $P$ is a point on the ellipse such that $|PF_1|:|PF_2|=2:1\;$, then how could I figure out of the area of $∆PF_1F_2$? As ...
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2answers
51 views

Proof that 4 points lie on a circle and that center of this circle lies on the circumcircle of $\triangle ABC$

Given is acute triangle $ABC$. Let $D$ be foot of altitude from vertex $A$. Let $D_1$ be a point so that line of symmetry between $D_1$ and $D$ is line $AB$. Let $D_2$ be a point so that line of ...
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0answers
125 views

Proove that α = η

The question is to show that α = η. The Angle QAP is the same as MAN, but AQ is not as long as AM and AP not as AN. I need help, I do not know how to solve it or how to start. (sorry for my bad ...
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2answers
64 views

Is it possible to find a closed form for $x$?

To solve the problem, I followed the following steps: Is it possible to find a closed form for $x$? $$\frac{\sin(x)}{\sin(\beta-x)}=\frac{\sin(\alpha)\,\sin(\theta-\gamma)}{\sin(\gamma)\,\sin(\...
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1answer
74 views

Right triangle geometry problem

Right triangle $\Delta ABC$ ($\angle ACB=90°$). The following is constructed: from point $C$ altitude $CD$, angle bisector $CL$ of $\angle ACB$, angle bisector $DK$ of $\angle ADC$, angle bisector $DN$...
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3answers
350 views

A curious geometry problem: Find the $\angle OBC$

I can not find a Method for to solve this geometry problem.I don't even know how to start.In fact, I didn't want to add my nonsensical attempts. I looked for a similar question to this question (...
4
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2answers
87 views

Prove that $BH=AH$

A triangle $ABC$ is given. There's a point $P$ inside it and also it is connected to point $H$, which lies on edge $BC$ ($H$ must not be the middle point of edge $BC$). Turns out, that bisector of ...
6
votes
1answer
101 views

Let $\Delta ABC$ be a right triangle. Prove that $\angle BEH=\angle HCI$.

Let $\Delta ABC$ is a right triangle. $D$ is chosen arbitrarily in $AB$,the segment $DH$ is perpendicular to the segment $BC$ at $H$, $E\in AC$ such that $DE=DH$. $I$ is the midpoint of $HE$. Prove ...
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1answer
26 views

Finding the location of points of a triangle given the angle and length ratio.

Given that a point P is located at (-2.5,4.33) I need to locate the points A and B such that $\frac{PA}{PB} = \frac{4.77}{8}$ and $\angle APB = 55^o $. A and B must be on the -ve part of the x axis. ...
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0answers
30 views

Plane Geometry problem: $a(PA)^2\ +\ b(PB)^2\ +\ c(PC)^2$ is minimum

Problem: Let $ABC$ be a fixed triangle and $P$ be a variable point in the plane of $\Delta ABC$. If $a(PA)^2\ +\ b(PB)^2\ +\ c(PC)^2$ is minimum, then the point P with respect to $\Delta ABC$ is (A) ...
0
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1answer
21 views

Know rotation with 3 coplanar points [duplicate]

I'm doing a computer vision program in which I have three coplanar points and I want to know their position at all times. In other words, I want to realize when there is translation and when there is ...
0
votes
1answer
29 views

Simple geometry question involving the circumradius of a triangle.

Prove that $\sin(2A)\overrightarrow{OA}+\sin(2B)\overrightarrow{OB}+\sin(2C)\overrightarrow{OC} =0 $ Where $O$ is the circumradius of the $ABC$ triangle How to approach this type of problem? How ...
0
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1answer
38 views

Find third triangle vertex given other 2 and lengths, without trigonometry

Given the coorinates of points A, B and lengths of all sides, point C should be found. I have a solution which relies on tangent equation and cosine rule $φ_1 = \arctan2(B_y - A_y, B_x - A_x)$ ...
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5answers
95 views

When did Pythagoras's formula for the hypotenuse change from $\sqrt{a^2 + b^2}$ to $\sqrt{a^2 + b^2 + 2ab}$?

I was in secondary school in Nigeria in the 60s during the transitioning from colony to independence to republic. At school we were given this formula that is now burnt into my synapses because our ...
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2answers
83 views

How to make correct system of equations to solve for the angles in this triangle?

I'm trying to solve this triangle for $X$. Thereby, I've tried to make correct system of equations. What would be the correct equations? Here are the equations I can find In $\triangle ABC$, ...
2
votes
3answers
33 views

Could you help me approach this problem using sine law?

Given that ABC is an isosceles triangle, $[BD]$ is angle bisector, $\angle BDA = 120^\circ$. Evaluate the degree of $\angle A $ Could you help me approach this problem using sine law? Here's my ...
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0answers
27 views

Feuerbach's Theorem in Spherical Triangle?

So first of all, just incase you don't know, Feuerbach's Theorem states the nine-point circle of a triangle in Euclidean space is tangent to the three excircles and the incircle of the triangle. ...
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0answers
26 views

Difference between SSA congurence rule and SAS congurence rule

if the bisector of any angle of triangle also bisect the opposite side then prove that triangle is an isosceles triangle. In the given problem when i try to prove congruence of two triangle i get ...
0
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2answers
43 views

Counting paths through a checkers board (only moving diagonally)

a) Imagine that a checker is placed in the bottom right corner of a 6 by 6 checker board. The piece mmay be moved one square at a time diagonally left or right to the next row up. Calculate the number ...