For questions about properties and applications of triangles

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Lemoine Point triangle

from Wolfram MathWorld, I know there is a Lemoine point of triangle, also called symmedian point, the sum of squared distances of this point to all the three sides is algebraically minimum. How can I ...
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4answers
46 views

In triangle ABC, Find $\tan(A)$.

In triangle ABC, if $(b+c)^2=a^2+16\triangle$, then find $\tan(A)$ . Where $\triangle$ is the area and a, b , c are the sides of the triangle. $\implies b^2+c^2-a^2=16\triangle-2bc$ In ...
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Intesection point of feet of altitudes

If triangle has vertexes at $(x_1,y_1),(x_2,y_2),(x_3,y_3)$, is the intersection points of feet of altitudes $$x_h = \frac{x_1x_2(y_2-y_1) + x_2x_3(y_3-y_2) + x_3x_1(y_1-y_3) + y_1^2(y_3-y_2) + ...
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1answer
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Finding the minimum value of squares of sides of a quadrilateral

What is the minimum value of $\frac{a^2+b^2+c^2}{d^2}$ where $a,b,c,d$ are the sides of quadrilateral I assumed the diagonals to be $p$ and $q$. I got that for minimum angle $A$ and $C$ must be ...
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Length of a right triangle created by skewing a rectangle's edge by a fixed amount

I have the above problem for a grid-based graphics system I'm working on, and I'm not sure if the math is solvable or not. I'm trying to determine the value of $A$. I've attempted to use ...
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1answer
63 views

Finding the value of k

If $x,y,z$ are perpendicular distances from circumcenter on the sides $BC,AC$ and $AB$ respectively. In need find $k$ such that $$\frac ax+\frac by+\frac cz=\frac{abc}{kxyz}$$ (Lowercase letters ...
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Finding the third side of a triangle, given ratio of two sides and difference of two angles [on hold]

Given $a=2b$ and $|\angle A-\angle B|=60$ degrees. Find the third side, where lowercase letters denote opposite sides and uppercase letter angles. Progress I could find the $\cos C$ but then ...
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Prove that $\frac{1}{2}ab \equiv \int_0^b \! f(x) \, \mathrm{d}x$ when calculating the area of a right triangle.

Triangle $ABC$ is a right triangle with sides $AB$, $BC$ and $AC$. $a$ is the length of $AB$. $b$ is the length of $BC$. $c$ is the length of $AC$. If $a = 3$, and $b = 4$, we can use ...
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Sum of segments inside a right triangle.

I am interested for a problem involving the sum of segments inside a right triangle. Consider a right triangle of hypotenuse $\overline{BC}$ and catheti $\overline{AB}$ and $\overline{AC}$. From the ...
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1answer
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Geometry Problem about tangent lines

Let S be the circumcenter of ABC. $A_0$ is the middle of arc BC not containing A, $C_0$ also the middle of arc AB without C. Let $S_1$ be a circle with center $A_0$, tangent to BC, $S_2$ with center ...
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1answer
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How to find mass points and ratios in a triangle?

How to find mass points with weights and ratios is my question. In my class, we learned about mass points. First we had the given ratios of 2 side lengths. Given: MC = d MB = e MA = f BD:DA = ...
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1answer
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An altitude is divided into 5 equal parts by 4 lines. Prove that the the areas of alternate sections are equal.

The question is as follows : Let their be a triangle ABC. Make altitude AD on C. Divide this altitude in 5 equal parts with lines EF, GM, IJ, KL intersecting at points M,N,O,P respectively. We have ...
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1answer
18 views

Trying to figure out coordinates of isoscleles triangle

I'm programming some application, which heavily depends of geometry. Let's say, in 2D coordinates system I have i.e. : Bxy = (5,-2) Cxy = (2,-5) ABlength = 5.5 ...
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Finding the distance between incenter and circumradius of a triangles [closed]

I need to show that the distance between inradius and circumradius is $(R^2-2Rr)^{1/2}$ where $R$ is circumradius and $r$ is inradius.
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1answer
53 views

In a Right Angled Triangle.

In a triangle ABC, Let $\angle$C=$\frac{\pi}{2}$. If $r$ is the inradius and $R$ is the circumradius, then what is the value of $2r+R$. Options are a+b b+c c+a a+b+c My approach. Radius of ...
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1answer
28 views

How is the area of this triangle calculated

I was reading "Problems of Calculus in one variable" by I A MARON, and came across this solved example in first chapter which I am unable to comprehend, please help me understand this. Scan of the ...
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1answer
34 views

Find cosine of acute angles in a right triangle.

If sides of a right triangle are in Geometric Progression, then find the cosines of acute angles of the triangle. [Answer] $\frac{\sqrt{5}-1}{2}$,$\sqrt\frac{\sqrt{5}-1}{2}$ My work: Using ...
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1answer
18 views

Two questions regarding the angle of reflection

I have two problems regarding the calculation of angles given certain values. In the first problem I need to calculate the angle X given that both angles Y are identical In the second problem I ...
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2answers
16 views

North has 0 degree and right angle has 90 degree although both are in same position

Before reading trigonometry I guessed that if a line is pointing to north then it has 0 degrees and increases clockwise. But now I see right angle has 90 degree though that is in the same position as ...
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25 views

How to find circumference origin position?

I need to find origin of circumference which is defined by two points and vertex angle of isosceles triangle: I've got radius of triangle by ...
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3answers
46 views

How was this equation for the hypotenuse of a triangle derived?

I've been staring at this for quite a while and simply can't understand how they got the equation for the hypotenuse. Probably has something to do with it being 5am my time! I'm confused because ...
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Finding the ratio of division by circumcenter

In an acute angled triangle ABC where O is the circumcenter Prove that $BD : DC = sin2C:sin2B \quad$ where D is the point of intersection of AO (extended) with BC. $AO : OD = sin2C + sin2B : ...
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using slope to find an angle in right angled triangle

I have a right angled triangle in which I know the length and the slope of the hypotenuse, how do I find one of the angles? Thanks
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$1,2,3$ task — calculate tangents

Given is square $ABCD$. Point $E$ is the midpoint of segment $CD$ ($E\in CD \wedge |DE|=|EC|$). Point $P$ is common point of diagonal $AC$ and line segment $BE$. ($\lbrace P\rbrace = AC \cap BE$). ...
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25 views

Coordinates of a vertex of a triangle?

Here is the problem: There is a triangle with vertices $A,B,C$ in a cartesian coordinate system, where coordinates of points $A$ and $B$ and the angle $\alpha=\measuredangle ABC$ are given. The ratio ...
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3answers
81 views

How many triangles in the picture?

Sorry if this has already been asked before. Is there any formula for such questions? EDIT: I have numbered the smallest triangles in the picture and marked the pentagon as x. Then I listed all ...
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2answers
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$A,B,C$ satisfy $\sin 2A: \sin 2B: \sin 2C= 5:12:13$ find $A$?

I would appreciate if somebody could help me with the following problem: Question: $A,B,C$ satisfy (1), (2) (1). $A+B+C=\pi(0< A,B,C< \pi)$ (2). $\sin 2A: \sin 2B: \sin 2C= 5:12:13$ Find $A$ ...
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31 views

Find length of side

I tried to solve this problem ... but i can't find answer. Anyone can help me? EBC=90 & DCB=90 & AHC=AHB=90
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1answer
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prove that $|x^2(y − z) + y^2(z − x) + z^2(x − y)| < xyz.$

If x, y, z are the sides of a triangle, then prove that $|x^2(y − z) + y^2(z − x) + z^2(x − y)| < xyz.$
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Area+Pythagoras Theorem

In the figure below show that Area A + Area B = Area C. What is the answer to this question?
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Proof related to circumcircle of triangle

I have a triangle $ABC$ with incenter $I$. $AI$ extended meets the circumcircle of $ABC$ at $M$. Prove that $CM=BM=IM$. I was able to prove that $CM=BM$ taking advantage of the fact that the ...
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3answers
80 views

Dividing an obtuse triangle into acute triangles

Can an obtuse triangle be subdivided into only acute triangles (right triangles are not allowed)? Any number of subdivisions can be made as long as all of the angles in all resulting triangles are ...
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1answer
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In this figure find AC=x

Can you find $AC$, when only the angle $DBC$ and $DEB$ are $90$ grades. I can't because I think they should give the angle $CAB=90$ grades too.
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Euclid I.24 Proof Why is DFG greater than EGF?

Proposition 24 If two triangles have two sides equal to two sides respectively, but have one of the angles contained by the equal straight lines greater than the other, then they also have the ...
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1answer
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To prove in a triangle: $AD^2=AB\cdot AC- BD\cdot CD$

If $AD$ is an angle bisector of $\triangle ABC$ (with $D\in BC$), then we have to prove that: $$AD^2=AB\cdot AC- BD\cdot CD$$ I have no idea how to do this, can this be proved with simple geometry? ...
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Lengths of the sides of a triangle: sufficient and necessary condition?

For any three positive scales, $a,b,c$, what is the sufficient and necessary condition such that they can form a triangle? Is $a+c>b,a+b>c,b+c>a$ enough? Thanks!
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Circles intersecting at A and B [duplicate]

Question: Two given circles intersect at A and B. A straight line through B meets the circles again at C and D. Prove that CD is greatest when it is parallel to the line joining the centres My ...
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178 views

Trigonometry. Finding the angle alpha

Refer the diagram below : What should be the angle alpha such that the variable x is between 7mm and 7.3mm.
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Isosceles triangle and scalene triangle

Question: Given the base and vertical angle of a triangle show that its area is greatest when the triangle is isosceles. My attempt: For isosceles triangle (with base given 2x, and vertical ...
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Can we find out the area of conical frustum by using triangles?

I have been trying to find out the area of conical frustum by using triangles.
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Does a triangle always have a point where each side subtends equal 120° angles?

Is there a point $O$ inside a triangle $\triangle ABC$ (any triangle) such that the angle $\angle{AOB} = \angle{BOC} = \angle{AOC}$? What do we call this point?
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Triangles with vertices on conics and their foci

Let $A$, $B$, and $C$ be the lengths of the three sides of a triangle. Let $α$, $β$, and $γ$ be the measures of the angles opposite those three sides respectively. Mollweide's formula tells us that ...
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1answer
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applied optimization problem- triangle fence

A farmer is trying to fence off a field on the edge of a river. He has two 1km long sections of fence to use to make a triangular field. The edge by the river does not need fencing, and the fence ...
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The height of a right triangle with legs $a,b$ is equal to $ab/\sqrt{a^2+b^2}$ [closed]

The height of a right triangle with legs $a,b$ is equal to $ab/\sqrt{a^2+b^2}$ Need help with number ii since it asks for a uncommon way of approaching the problem.
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Question - Corresponding parts of congruent triangles

Please answer the question below with these specifications: If the answer is yes write a paragraph proof to show which congruence shortcut utilized. Show all rules of geometry that are applied to ...
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1answer
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Is a triangle with two equal angles always isosceles?

An isosceles triangle is a triangle with two sides that are equal in length. This means that two angle will also be equal to each other. Is there any way that a triangle could have two (only two) ...
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1answer
28 views

Related rates question.

Two sides of a triangle have lengths $\sqrt{21}~m$ and $\sqrt{7}~m$. The angle between them is increasing at a rate of $\dfrac{2}{\sqrt{3}}~rad/sec$. How fast is the altitude of the triangle ...
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3answers
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Geometry question involving triangle

Question: $ABC$ is a right angle triangle at $A$. $AD$ is the altitude through A; E is a point on AC such that $AE=CD$. F is a poibnt on AB such that $AF=BD$. Prove that $BE=CF$. Challenge ...
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Getting 90 degree coordinate of 2 coordinates that you know

I have 2 coordinates and I need to find the third with a 90 degree angle. How could I do this? ...
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Geometry question involving triangles given with picture.

Here's the question: $\overset{\Delta}{ABC}$ is a triangle. $D$ is a point on $[BC]$. $|BD|=4$. $|AD|=|CD|$. $\text m(\widehat{CBA})=\alpha=30^\circ$. $\text ...