For questions about properties and applications of triangles

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1answer
66 views

A question on equilateral triangles

A point $D$ is on the side $BC$ of an equilateral $\triangle ABC$ such that $DC = \frac{1}{4} BC$ . Then $AD^2 = ?$ Image I drew... Options are $13 CD^2$, $9 AB^2$, $6 CD^2$, $12 ...
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2answers
49 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
39 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
166 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
94 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 ...
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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
73 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
44 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
151 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
130 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
58 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
41 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
81 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
144 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
136 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
56 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
337 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
532 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
1k 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
149 views

Triple systems with no six points carrying three triangles

Can anyone please send a link to this article? ...
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2answers
260 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 ?
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1answer
64 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
100 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
161 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
169 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
73 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
160 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
81 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 ...
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4answers
110 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
66 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
77 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
472 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
92 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 ...
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1answer
30 views

Figuring out the side of a triangle

I'm having trouble on this problem I don't know how to set it up. I know XO=2 and OB=6. I'd appreciate any hints.
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0answers
53 views

Triangle inscribed insemicircle area-ratio question

My approach: $m<A= 60$ degrees and $m<C=30$. This creats a 30, 60, 90 triangle with ratios $$1:\sqrt3:2$$ After getting the ratio's of the areas, I obtain $$\frac{b*h}{\pi r^2}=\frac{1*\sqrt ...
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1answer
59 views

height and bisector in a right triangle

We have right angle $ABC$ where $AB$ is hypotenuse, and angle bisector $CD$ we know that $AB=c$, $CD=u$. Express height depending on the $c, u$, and what is the condition between $c,u$ that the ...
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2answers
77 views

Number of Equilateral triangles in circle with 42 evenly spaced points?

I know that the answer is 42/3 = 14 points, or in general for a circle with N points it is N/3, but I don't know why it actually works. Why is the number of equilateral triangles for a circle with N ...
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1answer
158 views

Find Cathetus C1, C2 Knowing Hipotenuse or Find C1, C2, C3, C4 of Rectangle

I have a rectangle. I know all sides and 4 points for it (see black rectangle below). I resize one edge of this rectangle to any point (see resized red color rectangle and new point B). Here is the ...
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1answer
46 views

Computing a normal to triangle

The normal to surface is just a vector that is prependicular to the surface in particular point, right? So normale to the triangle is a vector perpendicular to his plane? I have a triangle, for ...
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1answer
125 views

Calculating radius of the circle

A perfect circle sits exactly within a right-angled triangle,dividing its hypotenuse into two segments of 3 and 10 units. The area of the triangle is 30 square units. What's the radius of the ...
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1answer
856 views

Is there a reflex angle symbol?

I only knew this angle symbol "$\angle$", which is usually used to represent acute angles. But now I have accounted a problem, where I wanted to represent the reflex symbol of $\angle ABC$. I know, I ...
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1answer
144 views

What is the shape of the region where a special point can exist?

For an equilateral triangle $ABC$ which has edge-length $1$, let $D,E,F$ be a point of contact of the inscribed circle and an edge $BC, CA, AB$ respectively. Then, let us call the region surrounded ...
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1answer
77 views

Heronian isosceles triangles

This is a problem from Project Euler, problem 94. The problem asks about isosceles triangles with integer sides (differing by 1 unit, e.g, 5-5-6) and integer area, which are known to be Heronian ...
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1answer
142 views

Obtuse-angled triangle equation

Give a obtuse-angled triangle and the obtuse angle is 105º. Find n such that the acute angles be the roots of the equation. ...
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3answers
76 views

Area of the triangle a,b,c

How to prove in any triangle that the area $X$ is given by: $$X=\frac{1}{4}(a+b+c)^2\tan \frac{A}{2} \tan \frac{B}{2} \tan \frac{C}{2}$$
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3answers
68 views

Prove that triangle is equilateral

I have a triangle here, how do I prove that $BCD$ is equilateral(so all lines have the same length) And yes this is 2D What I have so far is $$BAC = 120\deg$$ So how do I point out that $$BCD = ...