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Questions tagged [triangle]

For questions about properties and applications of triangles.

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2answers
16 views

$y=mx$ and $ax+by+c=0$ are perpendicular and meet at $(-9,6)$. Find the area of a related triangle.

I was trying to solve this problem, but I'm not sure how to approach this problem. Here is the problem: The lines $y = mx$ and $ax + by +c = 0$ are perpendicular to each other, and they intersect ...
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1answer
13 views

Joining walls into a trianglular prism

I'm helping to build a set for my son's theatre program, and am stumped by what is probably a simple trig problem (it's been a while). We want to build a triangular prism of walls, so that we can ...
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1answer
25 views

Given the coordinates of a box, how can I find the coordinates of a box that is X% bigger than the original box

So I'm given a box (the inner box in the picture) and I need to figure out the coordinates of the end points of the outer box given only the coordinates of the inner box. So far I've gotten the ...
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1answer
28 views

Calculate side lenght of triangle from two rectangle on top of each other

I want to calculate the side lenght of $b$. I have two rectangles with one at 0° (screen) and I have one rectangle at 20° (turned image). With respect to the middle point. Both rectangles have a ...
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1answer
25 views

Find complex number representing circumcentre and orthocentre of a triangle with vertices represented by complex number z1, z2, z3?

I am having hard time finding it, the method for finding orthocentre and circumcentre we use in coordinate geometry won't work here if complex number are not given in form of a+ib.
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2answers
91 views

How to use the Law of Sines to Find an Angle

I am trying to figure out how to find an angle with the law of sines. I have a triangle where: A = $120^\circ$ B = unmarked C = $\theta$ a = 45 b = unmarked c = 36 How can I find the angle ...
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1answer
29 views

prove that an isosceles triangle has the largest area

Inside this angle, alpha is place in a segment of length whose ends are on the sides of the angle so that the area of ABC was maximum
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1answer
26 views

circumcentre of equilateral triangle

Given the circumcenter of an equilateral triangle how can i find the length of the side of the triangle? The exact question is like this: An equilateral triangle with circumcenter at (-2, 5) having ...
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2answers
40 views

ABC triangle has vertices $A(3,5) B(-7,1)$ and $C(5,5)$. Write the equation of the median from vertex A. [on hold]

The answer is $A$: $x-2y+7$ according to the book, but I want to know how to solve it. Please show the solution step-by-step.
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1answer
52 views

An interesting (conjectural) property of any triangle

Given any triangle $\triangle ABC$, we can always build three ellipses, each of them having foci in two of the vertices and passing through the third one, as shown in the following picture: In ...
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1answer
95 views

Prove that $SD'$, $EF$and $HI$ are concurrent?

Let $\triangle ABC$ be a triangle with incenter $I$ and orthocenter $H$. The incircle of $\triangle ABC$ touches $BC$, $CA$, $AB$ at $D, E, F$ respectively. Let $D'$ be the reflection of $D$ through $...
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2answers
45 views

$AH=AS$ where $H$ is the orthocenter of $\triangle ABC$ and $S$ is the midpoint of the arc $BHC$ of the circumcircle of $\triangle BHC$

The altitudes of an acute triangle $ABC$ which is not isosceles concur at the point $H$. Let $S$ be the midpoint of the circular arc $BHC$ of the circumcenter of the triangle $BCH$. If $AS$ and $AH$ ...
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3answers
81 views

In a rhombus $ABCD$, prove that $IG\perp IP$. [closed]

Let $ABCD$ be a rhombus with $\angle ADC=60^\circ$ (picture in attach file). The points $E$, $F$, $G$, and $H$ are midpoints on sides $AB$, $DA$, $CD$, and $BC$, respectively. Let $J$ be the ...
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1answer
50 views

A conjecture involving three parabolas intrinsically bound to any triangle

Given any triangle $\triangle ABC$, we can build the parabola with directrix passing through the side $AB$ and focus in $C$. This curve intersects the other two sides in the points $D$ and $E$. ...
2
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1answer
73 views

An ellipse intrinsically bound to any triangle

Given any triangle $\triangle ABC$, we build the hyperbole with foci in $A$ and $B$ and passing through $C$. The hyperbole always intersects the side of the triangle that is opposite to the vertex ...
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2answers
132 views

A conjecture about the intersections of three hyperboles related to any triangle

Given any triangle $\triangle ABC$, we build the hyperbole with foci in $A$ and $B$ and passing through $C$. Similarly, we can build other two hyperboles, one with foci in $A$ and $C$ and passing ...
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4answers
46 views

Maximize area of triangle

I'm trying to maximize the area of a triangle with the following three sides. The first side is the y=0, the second lies on the line y = 3x, and the third passes through the point (1,1). I want to ...
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2answers
39 views

The area of a triangle with sides $11$, $12$, and $x$ is $28.9$. Find $x$.

I’ve been stuck on this question for days now, and honestly can’t figure it out. It’s not isosceles or right angled, and there’s no angles so I can’t use the sine or cosine rule. Any ideas? The ...
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2answers
15 views

Right angle triangle, all three angles known and two shorter sides add to equal 1 [closed]

So I have a right triangle with 90, 60, 30 degree corners. The two shorter sides add to equal 1. (a+b=1) The question is to find the longest side (c). The given answer states that the longest side ...
3
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2answers
83 views

Maximize the value of $\sqrt{x-x^2}+\sqrt{cx-x^2}$ without using calculus

Assume that $c$ is positive. How can we maximize the value of $\sqrt{x-x^2}+\sqrt{cx-x^2}$ with respect to $x$ without the use of calculus? With calculus, we can easily find out that the max of the ...
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1answer
75 views

If $\sin\alpha+\sin\beta+\sin\gamma=\cos\alpha+\cos\beta+\cos\gamma=0$, then find the values of $\sum \sin^2 \alpha$ and $\sum \cos^2 \alpha$.

If $\sin \alpha+\sin \beta+\sin \gamma=0$ and $\cos \alpha+\cos \beta+\cos \gamma=0$ Then find the values of $\sum \sin^2 \alpha$ and $\sum \cos^2 \alpha$. Try: $$(\sin \alpha+\sin \beta+\sin \...
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1answer
24 views

Area of the shaded section involving triangles in a square

In the image, $ABCD$ is a square of side $4$. $\triangle DFC$ is equilateral, then the shaded area is? My try was extending $DF$ and $CF$, meeting $AB$ and then i tried using the $30-60-90$ triangle ...
2
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2answers
71 views

Find an angle in a geometric figure (given) considering triangles

Question: In the figure below, AC=AB and AD=BC. Find angle $x$. My attempt: using a geometric approach, consider the following figure (proportions are not exact). Using the notation $AC=AB=b$ and $...
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0answers
22 views

pedal circle passes through the anti steiner point?

Given a triangle $\triangle ABC$ and $P$ is an arbitrary point. Let $\triangle DEF$ be the cevian triangle of $P$ with respect to (wrt) $\triangle ABC$ and $\triangle XYZ$ be the pedal triangle of $P$ ...
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0answers
22 views

Inscribed circle with concurrent chords

(Inspired by this problem) In $\triangle ABC$ let the incircle touch $BC$, $CA$ and $AB$ at $D$, $E$ and $F$ respectively. Let line $AD$ cut an incircle at point $L$ and line $LB$ and $LC$ cut ...
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1answer
36 views

Easy proof of the intercept theorem.

Im looking for a simple proof of the Intercept-Theorem in the Euclidean Plane $\mathbb{R}^2$. I can use analytic and synthetic Proofs and Theorems but students should be able to understand it. I've ...
2
votes
2answers
54 views

Show that $A^2 +B^2+C^2=D^2$ using the following diagram (tetrahedron)

Slicing a corner off a square gives a right-angled triangle, as shown in the diagram below. The lengths of the sides of this triangle are related by Pythagoras’s theorem: $a^ 2 + b^ 2 = c^ 2$ . Show ...
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1answer
20 views

the number of such point

$P(3,1), Q(6,5)$ and $R(x,y)$ are three points such that the angle PRQ is a right angle and the area of triangle $RPQ = 7$ , then what are the number of such point R ? My Try : enter image ...
2
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1answer
78 views

Distance between the Orthocenter and the Lemoine Point (Symmedian Point)

I need help to solve this problem. I have no idea how to solve. Is there previously solved copy of this problem? Thank you very much. Let $H$ be the orthocenter of a triangle $ABC$ with ...
4
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2answers
95 views

Is there an elementary method of finding this missing angle?

Let a point $P$ lie in a triangle $\triangle ABC$ such that $\angle BCP = \angle PCA = 13^\circ$, $\angle CAP = 30^\circ$, and $\angle BAP = 73^\circ$. Compute $\angle BPC$. I have an ugly trig ...
2
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1answer
38 views

perfect right-triangles that are not super-perfect

I'm seeking some insight on the answer to this problem from Project Euler. Consider the right angled triangle with sides $a=7, b=24$, and $c=25$. The area of this triangle is $84$, which is ...
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0answers
51 views

how to get the center of moved equilateral triangle according to endpoints displacement?

sorry if I did not use the proper jargon because I can't recall any specific words. $\mathbf Conditions:$ There is an equilateral $\Delta ABC$ in $\Bbb{R^3}$ with given side-length which lies on $...
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2answers
32 views

Geometry: Proof related to right angled isosceles triangle

ABC is a right angled isosceles triangle. If AD is a bisector of angle BAC then prove that AC + CD = AB. The right angled isosceles triangle The right angle is at C.
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0answers
12 views

Show that the sum of inradii and exradii of $\Delta HX_iX_j$ where $X_k \in \{A,B,C\}$ of $\Delta ABC$ and its orthocentre $H$.

In an acute $\triangle ABC$, denote by $r_1,r_2,r_3$ the exradii and $k_1,k_2,k_3$ denote the respective inradii of $\triangle HBC, \triangle HCA, \triangle HAB$, then show that $r_1+r_2+r_3+k_1+k_2+...
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1answer
74 views

Proving that one of the angles is twice the other

I have the following question with me: "Let ABC be an isosceles triangle with $AB=AC$. Let $D$ be a point on $BC$ such that $BD = 2DC$ and let $P$ be a point on $AD$ such that angle $BAC$ = angle $...
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1answer
29 views

Problem with 2 similar right-angled triangles [closed]

There are two similar right-angled triangles, where known information is only one side from one, another side from the other triangle, they are not hypotenuse and right angles. Is it possible to ...
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2answers
24 views

Find height of a wall if at the beginning it exceeds $10$ meters and then $8$ meters

When the foot of a staircase is $5$ meters from the base of a wall, it protrudes $10$ meters above the wall; and if it is $9$ meters from the base, it stands $8$ meters. Find the height of the wall. ...
1
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1answer
24 views

Ascending triangle wave formula

I'm trying to find a formula which can plot points that move down one unit 60% of the time, and up two units 40% of the time, similar to how the stock markets move, in an up trend. I started with the ...
3
votes
3answers
40 views

Right angles in triangles formed in trapezoid

In the following trapezoid, is angle A in triangle ABC a right angle even though it isn't labeled as such? And if so what property would we use to determine that? I was able to get the correct area ...
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1answer
30 views

Distance length and viewing angle

I'm developing a game through BabylonJS and there's four values that are of interest. The environment around the person is a sphere. This sphere has radius 100. The person may move forward from the ...
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2answers
32 views

Constant area ratio of circles and regular polygons

Consider these two shapes using squares and circles: On the left: there are two squares, one inscribed and one circumscribed in the same circle. The ratio of the two square areas is $2$: $$ \cfrac{...
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1answer
60 views

If $\triangle DEF$ is inscribed in $\triangle ABC$ such that three subtriangles have equal inradii, find inradius of $\triangle DEF$

I have a triangle $ABC$ with inradius $r$. Points $D$, $E$, $F$ are chosen on side $BC$, $CA$, $AB$, respectively, such that $\triangle AFE$, $\triangle BDF$, and $\triangle CED$ have same inradius $...
2
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2answers
113 views

Help in proving that the given triangle is isosceles

I have the following question with me: Let $ABCD$ be an isosceles trapezium with $AB$ parallel to $CD$. The inscribed circle of triangle $BCD$ meets $CD$ at $E$. Let $F$ be a point on the internal ...
3
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5answers
158 views

Find the relationship of the length of triangle's sides.

Denote the three sides of $\triangle ABC$ to be $a,b,c$. And they satisfy $$a^2+b+|\sqrt{c-1}-2|=10a+2\sqrt{b-4}-22 $$ Now determine what kind of triangle $\triangle ABC$ is. A.Isosceles triangle ...
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1answer
47 views

Proving that the given triangle is isosceles

I have the following geometry problem with me: Let $O$ and $O_1$ be the centres of the incircle and excircle opposite to $A$ of triangle $ABC$. The Perpendicular bisector of $OO_1$ meets lines $AB$ ...
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1answer
27 views

Why is the ratio of any two sides of an equilateral triangle on a complex plane equal to a complex cubic root of unity?

Why is the ratio of any two sides of an equilateral triangle on a complex plane equal to a complex root of unity? This question is derived from an excerpt from the following note, which states that: ...
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0answers
38 views

mimimum area of inscribed isosceles right angle

Problem: A(-3,0), B(2,0), C(0,3) P, Q, R is on AB, BC, CA respectively. AP:PB=t:1-t , BQ:CQ=u:1-u, CR:RA=v:1-v triangle PQR is Isosceles triangle such that angle(PRQ)=90deg When area of PQR is ...
2
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4answers
84 views

graphical representation of trig functions

I'm currently learning the unit circle definition of trigonometry. I have seen a graphical representation of all the trig functions at khan academy. I understand how to calculate all the trig ...
0
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1answer
62 views

Incenter geometry problem

Let $I$ be the incenter of $\triangle ABC$ and let $\overline{AI}$ meet the circumcircle of $\triangle ABC$ at $D$. Denote the feet of perpendicular from $I$ to $\overline{BD}$ and $\overline{CD}$ by $...
0
votes
1answer
42 views

Problem involving the incentres of the triangle formed by drawing the altitude of the triangle

Let ABC be a triangle and D be the foot of perpendicular from A onto BC. Let E and F be incentres of triangles ABD and ADC respectively. The line EF when extended both sides meet AB in K and AC in L. ...