For questions about properties and applications of triangles

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2answers
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Question on inscribed equilateral triangle

Question: $ABC$ and $ODE$ are equilateral triangle with $BC || DE$. If $O$ is the center of the circle, then find the ratio $AQ:QC$ So, my thought on this is that, since we are not given the ...
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3answers
563 views

Find radius of a circle which is tangent to three known lines

I need to find the equation for a circle which is tangent to the following three lines: y=0 x=0 y=-x+0.338334 For the last tangent line equation, I know that it is tangent at the point (0.169167, ...
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4answers
229 views

Using the Law of Sines to find all triangles with given values of two sides and an angle

Our teacher skimmed over this and we have homework over it. Textbook is mostly unhelpful. I'm confused on how ambiguous case works, and everything I see online just confuses me more. I'm not quite ...
2
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1answer
658 views

equality of triangle inequality

$z$ and $w$ be nonzero complex numbers. How do I show that $|z+w|=|z|+|w|$ if and only if $z=sw$ for some real positive number $s$. I approached this by letting $z=a+ib$, and $w=c+id$, and kinda ...
3
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1answer
122 views

slice up a slice of a triangle into n areas of equal size

Figure description: The point $(0, 0)$ is in the upper left corner. The coordinate system grows to the lower right corner. The short sides of the big triangle have the same length. I want to slice ...
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2answers
66 views

Formula for sides of a triangle where the Perimeter equals to the Area [duplicate]

I was wondering if there is a formula that could generate the values of the sides of a triangle where his area equals to his perimeter. I only found that if the triangle is equilateral then ...
2
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1answer
439 views

Angle between a plane defined by three points (x, y, z are unknown) and the horizontal

I am a novice in mathematics, and I have a question: Suppose that I have 3 points in the space: (x,y,z) for these points are not known for me. given that I know the angles a, b and c (c.f. above ...
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1answer
667 views

Proving triangles congruent with circles

I have a proof that looks like the following, not really sure where to start/how to solve. Any help would be appreciated. Given: circle $S$ and circle $T$ intersect at $M$ and $O$. Prove: $\triangle ...
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2answers
4k views

Cross section with equilateral triangles and integration

Hello guys so I needed help with a problem which is: Let $S$ be the solid with flat base, whose base is the region in the $xy$-plane defined by the curves $y=e^x$, $y=−2$, $x=1$ and $x=3$, and ...
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2answers
171 views

Show that the triangle which satisfy the inequality $\frac{\sin^2 A+\sin^2 B+\sin^2 C}{\cos^2 A+\cos^2 B+\cos^2 C}=2$

Show that the triangle which satisfy the inequality $\dfrac{\sin^2 A+\sin^2 B+\sin^2 C}{\cos^2 A+\cos^2 B+\cos^2 C}=2$ is right angled. My work: $\sin^2 A+\sin^2 B+\sin^2 C=2(\cos^2 A+\cos^2 ...
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1answer
145 views

If $A,B,C$ are the angle of a triangle, then show that $\sin A+\sin B-\cos C\le \dfrac3 2$

If $A,B,C$ are the angle of a triangle, then show that $\sin A+\sin B-\cos C\le \dfrac32$ I tried substituting $C=180^\circ-(A+B)$ and got stuck. I also tried using the formula $\sin A+\sin B=2\sin ...
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1answer
41 views

Simple Question on Triangles…

What times the sum of the squares of the sides of a triangle is equal to the sum of the squares of the medians of the triangle.
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1answer
55 views

A question on similar triangles…

In the given figure, AB, EF and CD are parallel lines. Given that EG = 5 cm, GC = 10 cm and DC = 18 cm, then EF = ??
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1answer
56 views

A question on equilateral triangles

A point D is on the side BC of an eqilateral triange ABC such that DC = 1/4 BC . Then AD^2 = ??? Image I drew... Options are 13 CD^2 , 9 AB^2 , 6 CD^2 , 12 BC^2 .
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2answers
47 views

magical isoceles triangle and 13/15 ratio

It seems eerily magical that $\dfrac {13}{15}$ corresponds within $99.926$ percent accuracy to the height of an isoceles triangle the height of isoceles $= \sqrt {.75} = 0.8660254...$ and $\dfrac ...
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1answer
36 views

Geometry question (finding the coordinates of a point)

Find the coordinates of a point on the y-axis which is equidistant from A(1,-3,7) and B(5,7,-5) I understand an isosceles triangle can be formed from these points, and the x,z coordinates of this ...
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0answers
158 views

History of incenter and Euler line

It is easy to see that if a triangle is isosceles, then its incenter lies on its Euler line. Who first proved the converse of this result and what technique was used? (See the post "The incenter and ...
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1answer
90 views

Find the area of the Grayed triangle Given the following Figure

Can you help me find the area of the gray triangle in the given figure. I'm having a hard time finding the base value of the triangle, I've managed to find the sides for the big triangle but not ...
3
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1answer
1k views

Why is it called an “orthocenter”? What is the orthocircle?

We know that the orthocenter of a triangle is the place where the triangle's three altitudes intersect. But why is it called that? The *in*center - the intersection of the triangle's three angle ...
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1answer
69 views

Find rotation angle of given image

At first: our aim is to find the total transformation of left house to the right house. What I did it first is translating the house with the center to the origin. I already found out that the ...
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4answers
42 views

If $a$ and $b$ are cathets a right triangle whose hypotenuse is $1$ determine the highest value of $2a + b$

Can some one help me out on where to go? If $a$ and $b$ are cathets a right triangle whose hypotenuse is $1$ determine the highest value of $2a + b$ ?
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0answers
135 views

Given 3 Vertices of a Tetrahedron, Find the 4th

A regular tetrahedron is circumscribed by the Earth (assume spherical). You are given 3 of the 4 vertices (as latitude and longitude in decimal format), and asked to find the 4th. Any help is most ...
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3answers
1k views

How many triangles can be created from a grid of certain dimensions?

How would you determine how many non-degenerate triangles can be drawn by connecting points in a $5 \times 5$ grid?
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2answers
108 views

Property of angle bisectors in a triangle

Let $ABC$ be a triangle having circumcenter $O$. Suppose $AH$ is the altitude from vertex $A$ and $AT$ bisects angle $A$. I would like a simple geometric proof that $AT$ also bisects angle $OAH$.
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2answers
52 views

Calculating the length of a line in a triangle

I feel very stupid, but I have to answer this question but I cannot seem to solve it! :( I have to find the length of DF. I already figured out that because angle C = angle A1 (left part of the ...
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1answer
38 views

Characteristics emerging from subdividing an obtuse scalene triangle?

I'm relying only on the geometry I learned in high school. Given a scalene obtuse triangle $ABC$, where $AC$ is opposite the obtuse angle, and a point $D$ in $AC$ such that $AD = DC$ (a midpoint). ...
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1answer
75 views

How to compute the *vertical* distance between a point and a triangle in 3D?

The point is either above or under the triangle i.e. if you project the point and the triangle on the ground, the point lies in the triangle. I want the distance DD' (in dark red) on the Z axis of ...
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1answer
126 views

Finding angle on an inclined plane

How can I go about finding the angle, theta, in this Physics problem? As you can tell, the right-most triangle is a simple 30-60-90 triangle, so above the right angle is a 60deg angle. Then the ...
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1answer
32 views

what $h_a$ in this triangle question.

Square $PQRS$ is inscribed into $\triangle ABC$ so that vertices $P$ and $Q$ lie on sides $AB$ and $AC$ and vertices $R$ and $S$ lie on $BC$. Express the length of the square’s side through $a$ and ...
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1answer
117 views

Completing a very difficult triangle

I have an isosceles triangle with the two equal sides of length 'c', and the bottom of length 'a'. Both base angles of the triangle have measures of 'a', in degrees. For example, if 'a' were 50, both ...
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1answer
44 views

liquid triangles

note: for simplicity the 2-dimensional case is described here. a similar situation could be treated in higher dimensions. liquids are distinguished by their ability to change form whilst retaining ...
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2answers
55 views

Relation between the radius and the area of tangential polygon

I've recently found a book with loads of formulas for triangle area, but unfortunaly the formulas were just listed, there wasn't a proof for them. I've tried to proof them. But I've stopped at one of ...
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5answers
293 views

Mensuration question

I recently came across a puzzling question: Two rectangles ABCD and DBEF are as shown in the figure. The area of DBEF is: Figure (hand-made): I know that through Pythagoras, we get ...
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1answer
494 views

Maximum area of a rectangle inside a triangle

I recently came across a problem where it gave a triangle with integer side lengths, and it asked you to find the maximum area of a rectangle of a triangle. I solved the problem correctly, but it ...
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1answer
78 views

Angle bisector in a triangle

For the angle bisector $I_a$ in a triangle $ABC$ it holds $$I_a^2 = \frac{bc}{(b+c)^2}[(b+c)^2 - a^2]$$ If $I$ is the incenter, I wonder if there exist similar formula for the part $AI^2$.
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2answers
907 views

how many rectangles in this shape

I've learned in my high school the solution to such riddle: How many rectangles are there in this shape: the solution is through combinations: in this shape is a $5\times 6$ grid so the number of ...
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1answer
128 views

Triple systems with no six points carrying three triangles

Can anyone please send a link to this article? ...
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2answers
248 views

Find value of the angle x

Find the value of the angle x. Plus : Someone could recommend me some good book about this subject ?
3
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1answer
63 views

Formula in a triangle

Let $H$ be the orthocenter in a triangle with sides $a, b, c$. Is it true that $$a^2 + HA^2 = 4R^2$$ where $R$ is the circumradius?
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1answer
99 views

Inequality in a triangle

Let $O$ be the circumcenter and $H$ the orthocenter in a triangle with sides $a, b, c$. Is it true that $$aOA^2+bOB^2+cOC^2 \ge aHA^2 + bHB^2 + cHC^2$$ or equivalently $$(a+b+c)R^2 \ge aHA^2 + bHB^2 + ...
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2answers
158 views

How prove this stronger than Weitzenbock's inequality:$(ab+bc+ac)(a+b+c)^2\ge 12\sqrt{3}\cdot S\cdot(a^2+b^2+c^2)$

In $\Delta ABC$,$$AB=c,BC=a,AC=b,S_{ABC}=S$$ show that $$(ab+bc+ac)(a+b+c)^2\ge 12\sqrt{3}\cdot S\cdot(a^2+b^2+c^2)$$ I know this Weitzenböck's_inequality $$a^2+b^2+c^2\ge 4\sqrt{3}S$$ But my ...
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3answers
162 views

Construct triangle given inradius and circumradius

If we know the inradius $r$ of a triangle and the circumradius $R$ we can find out the distance between the incircle $I$ and the circumcircle $O$: $OI^2 = R^2-2Rr$. Therefore we can draw the incircle ...
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1answer
70 views

A Pythagorean problem

We have two points F1, F2. F1-F2 is 21m. We have a point (P) outside the line. The line from F1-P is called D1. The line from F2-P is called D2. P is 12m away from F1-F2 on a straight line crossing ...
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2answers
140 views

Area of an equilateral triangle divided by three lines

An equilateral triangle is divided by three straight lines into seven regions whose areas are shown in the image below. Find the area of the triangle. How to solve this problem ?
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1answer
77 views

construct an equilateral triangle with out knowing its scale

How do i construct an arbitrary equilateral triangle with out knowing its scale? for e.g. pick two points a and b. make $60$ degree acute angles at point $a$ and point $b$ and the two angles meet at ...
3
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4answers
107 views

How to prove that triangle inscribed in another triangle (were both have one shared side) have lower perimeter?

This question looks really simple, but to my (and my co-workers) frustration we were not able to prove this in any way. I tried all triangle formulas known to me but I feel I'm missing the point, and ...
3
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1answer
62 views

Prove: Exactly a quarter of 3-part partitions of numbers >2 equal to 0, 2, 10 mod 12 will make a triangle.

Consider perimeters $>2$ equal to $0$, $2$, or $10 \mod(12)$. The sequence starts $10, 12, 14, 22, 24, 26, 34, 36, 38, 46, 48, ...$ and we can look at the three part partitions that make triangles. ...
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1answer
75 views

Which theorem should be used to solve this question?

My friend sent me that question and said "another Carnot theorem" is used to solve this question but i couldnt find that theorem. Can you help me? Additional explanation: $$ \widehat{ABD} = ...
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3answers
386 views

If two sides of a triangle are equal, and the angle between them is $60^\circ$, prove the third side is equal to the first two sides.

In other words, given points $A$ and $X$. Rotate $X$ $\,-60^\circ$ around $A$ to get point $X'$. How would you prove $XX' = AX = AX'$? I know this is true.
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1answer
91 views

explaining the resriction $b<a<2b$ in a triangle

I saw in a book that if $ABC$ is an isosceles triangle $(AB=AC)$ and the triangle is tangent to a circle in points $D,C$ and $AC$ is intersecting the circle in point $E$; $AC=a$, $BC=b$ so it has ...