# Tagged Questions

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### How to get the third point coordinates in isosceles triangle?

Isosceles triangle $ABC$ $AB = AC = d_1$ $BC = d_2$ $A = (x_1, y_1)$ $B = (x_2, y_2)$ $C = (x_3, y_3)$ $\angle BAC = \phi$ $\angle ABC =\angle ACB = \theta$ I want an equation for $x_3$ and $y_3$ ...
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### Number of triangles formed by all chords between $n$ points on a circle

We have $n$ point on circumference of a circle. We draw all chords between this points. No three chords are concurrent. How many triangles exist that their apexes could be on circumference of ...
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### problem about length of perpendicular chords

Question $AB$ is chord of circle $O$,points $D$ and $E$ are chosen on $AB$ in a way that $AD=BE$.prove two chords that are perpendicular to $AB$ and pass $D$ and $E$ points are equal.(prove $LK=MN$) ...
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### Triangle Theorem relating the shortest and longest distance from any arbitrary point inside

I recall somewhere there was a relationship such that given a triangle T and a point A: if A is inside of T, then the sum of the longest distance from A to any point on a side of T, plus the shortest ...
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### Circle with perpendicular chords

A blue circle is divided into $100$ arcs by $100$ red points such that the lengths of the arcs are the positive integers from $1$ to $100$ in an arbitrary order. Prove that there exists two ...
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### Minimum Value of $x_1+x_2+x_3$

For an Acute Triangle $\Delta ABC$ \begin{align}x_n=2^{n-3}\left(\cos^nA+\cos^nB+\cos^nC\right)+\cos A\,\cos B\,\cos C\end{align} Then find the least value of $$x_1+x_2+x_3$$ My Approach: I have ...
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### Rotation matrix of triangle in 3D

How can I find out the rotation matrix of a right angle triangle defined by 3 points in 3D space (assuming the un-rotated triangle faces the x axis)
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### Area of a triangle whose each side is less than 2 and greater than1.

What is the area of a triangle if each of its sides is greater than 1 and less than 2? My Try:Let a,b,c be the sides of triangle,then ...
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### Similar triangle, Quick question (Thick Lens Formula)

http://www.panohelp.com/thinlensformula.html On the right hand side, f is defined as focus of the lens, i understand why the image distance is (f + fm). However i have spent an afternoon and could ...
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### If you know 2 sides of the triangle, wha is the third side?

I understand why A & C are correct but I don't get how E is a possible length since whatever number I plug in for x I get a number greater than 5x+5...
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### How many pieces of information are needed to determine a triangle?

Typically 2 sides and 1 angle need to be given in order to determine a unique triangle. Alternatively 1 side and 2 angles, or the Cartesian coordinates of three vertices, or the area, base, and ...
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### Triangle problem - finding the angle

Please look at the following figure: All the angles are in degrees. I have to find $x$. I am really no good at solving geometry problems. I tried to search the internet for similar problems and ...
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### Tangent and angle bisectors

The tangent to the incircle of a triangle ABC is reflected about the external angle bisectors. Show that the triangle formed by the resulting 3 lines is congruent to ABC .
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### Locus of the Orthocenter of the Traingle

Coordinates of $\Delta ABC$ are $A(3,4)$, $B(5 \cos\theta, 5 \sin\theta)$ and $C(5 \sin\theta,-5 \cos\theta)$. Find the locus of its orthocenter. My idea: It is clear that $(0,0)$ is equidistant ...
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### Geometric proof with a isosceles triangle

Given is $\triangle ABC$ with the medians $AD$, $BE$ with $|AD|=|BE|$. The medians intersect in $S$. a. Use similar triangles to show that $|AS|:|SD|=|BS|:|SE|=2:1$. b. Prove that $\triangle ABC$ is ...
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### Paths followed by Morley triangle vertices as apex moves parallel to base

Let the vertices of a triangle $T$ be $(A,B,C)$, and $(a,b,c)$ the vertices of its Morley triangle $M$. Designate vertex $C$ as the apex of $T$. Now move apex $C$ parallel to $AB$, all the while ...
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### Triangle in 3D space point X and Y coordinate know find Z

I have a triangle in a 3D space. I know the points X an Y coordinate but I dont know the Z. How can the Z be calculated by knowing the points of the triangle and the X an Y coordinate of the point ...
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### Finding an area of a triangle inside of a triangle, given certain areas of other triangles, and area ratios.

I'm studying for the Waterloo Math Contest (Galois, Gr. 10) taking place in April of 2015 and I am preparing by looking at previous problems and solving them. This is question 4(c) on the 2010 Galois ...
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### Get the angle in a circle using center, radius and one point in a circle.

There is a circle and i know Point1 this is fixed and i know another point Point2 which can be anywhere in the circle. and i want to know the angle which is made at center. Thanks Your help will be ...
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### How to calculate a variable vertex's coordinates on a scalene triangle given an original triangle

The vertex I'm looking for lies on one of the altitudes of the red triangle which we know everything about via calculation. Given the desired, final angle (135 degrees, but theoretically, any ...
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### Find length of $CD$ where $\angle BCA=120^\circ$ and $CD$ is the bisector of $\angle BCA$ meeting $AB$ at $D$

$ABC$ is a triangle with $BC=a,CA=b$ and $\angle BCA=120^\circ$. $CD$ is the bisector of $\angle BCA$ meeting $AB$ at $D$. Then the length of $CD$ is ____ ? A)$\frac{a+b}{4}$ B)$\frac{ab}{a+b}$ ...
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### Finding the area of a triangle in terms of the radius of the excircle

Prove that the area of a triangle $ABC$ is $$\frac12 (b + c - a)r$$ where $r$ radius of the excircle opposite to $A$ and the rest of the symbols have their usual meaning. I started off with the ...
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### Rationalizing triangle relationship for a bar inside a hemisphere

Find ratio of length AE to diameter of the hemisphere Given that ABD = 90 degree, AO = BO, O is the center of the sphere. This is actually a physics problem, but I bump into geometric problem ...
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### Rationality in Triangle

How can I justify this answer? I think the answer is infinite, but cannot justify it///
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### Circle theorem/triange angle question

I am doing practise papers and there is one question I cannot understand even with the mark scheme. I have added the pictures below: Question (with added annotations): Mark scheme: The question ...
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### Hyperbolic Triangles and Uniform thinness

My textbook states that all triangles in hyperbolic space are uniformly thin in the following way: If $ABC$ is a triangle and $x$ is a point on one side, then there exists a point $y$ on one of the ...
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### Length of a line in an isosceles triangle. (mind boggling )

In an isosceles triangle ABC, side AB and AC are equal in length. There exists a point D on the side AB. The angle BAC is theeta . The side AD is two units smaller than AC .What is the generalized ...
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### Maximal area covered by two triangles in unit circle

What is the maximal area covered by two triangles in a unit circle? There are no restrictions other than that. They can overlap, touch the circle, not touch the circle etc. So far I have shown In ...
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### Closest Points on Two Triangles in 3D Space

I have two triangles in 3D space, defined by 3 (x, y, z) points each. I'm looking to find the closest points between the two triangles, whether that be on surface, edge, or point. I'm unsure how to ...
Problem: (Euclid had a triangle in mind - I am including this line so that future googles come across this question) The triangles longest side is $20$ and another side is $10$. Its area is $80$. ...
Problem is to find the ratio of the area of the circle to that of the semi-circle. Note that points $F$ and $E$ weren't given in the original diagram, and that the circle at the top-right ...