-1
votes
1answer
42 views

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 $y_3$ ...
0
votes
1answer
39 views

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 ...
1
vote
1answer
41 views

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 $LK=MN$) ...
0
votes
1answer
16 views

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 ...
5
votes
1answer
48 views

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 ...
1
vote
1answer
89 views

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 ...
0
votes
1answer
32 views

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)
2
votes
1answer
43 views

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 ...
0
votes
0answers
28 views

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 ...
0
votes
1answer
53 views

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...
2
votes
4answers
65 views

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 ...
3
votes
2answers
94 views

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 ...
9
votes
5answers
266 views

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 .
0
votes
2answers
36 views

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 ...
0
votes
1answer
26 views

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 ...
3
votes
1answer
47 views

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 ...
0
votes
1answer
38 views

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 ...
2
votes
3answers
69 views

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 ...
2
votes
2answers
32 views

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 ...
1
vote
1answer
70 views

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 ...
1
vote
2answers
25 views

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)$\frac{ab}{a+b}$ ...
1
vote
1answer
35 views

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 ...
0
votes
0answers
17 views

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 ...
1
vote
0answers
23 views

area of triangle in terms of sides ratio [duplicate]

In triangle $ABC$, $X$ and $Y$ are points on the sides $AC$ and $BC$ respectively . If $Z$ is on the segment $XY$ such that $AX/XC=CY/YB=XZ/ZY$, prove that the area of triangle $ABC$ is given by: $$ ...
1
vote
2answers
41 views

Proving in a triangle

$AB$ and $CD$ are two straight lines intersecting in $O$. $XY$ is another straight line. Show that in general two points can be found on $XY$ which are equidistant from $AB$ and $CD$. But isn't ...
0
votes
1answer
29 views

Juxtapose two triangles with a common edge

I'm not experto in geometry but I'm trying to do a software that handle triangles in various way. And I'm trying to learn geometry, of course : ) I have one fixed triangles $T1 = \hat{ABC}$ and a ...
0
votes
0answers
50 views

Finding general Cartesian coordinates of the third vertex of a triangle lies between two circles

I'd like to find the Cartesian coordinates of the vertices $(a, b, c)$ of the triangle $T$ inscribed in the circle $S^1$ and circumscribed about the circle $D$ ? I start my calculations as follows: ...
0
votes
2answers
37 views

Geometry, two perpendicular lines

"Let $\hat{ABC}$ be an isosceles triangle with $AB=AC$. $D$ is a point on $BC$ such that $DC=DB$ (middle of $BC$). $E$ is the projection of $D$ on $AC$ and $F$ the middle of $DE$. Prove, using vectors ...
0
votes
0answers
37 views

minimum sum of distances from vertices

Find a point on the plane of a triangle such that the sum of its distances from three vertices is minimum. I am guessing that it is the centroid but I can't prove that.
0
votes
2answers
25 views

Can the vertex angle of an isosceles triangle be found without the law of cosines (no calculator)?

If we know three sides of an isosceles triangle, can we find the measure of the angles without using a calculator (that means no law of Cosines/Sines).
0
votes
1answer
19 views

Finding a side of a triangle with one side, angle and a quotient of two other sides.

Solving a firing-with-prediction puzzle in the game I am developing I found myself looking on the internet of solutions about triangle and its side. It turns out there is very few information about ...
5
votes
3answers
225 views

Need algebra tip about $a^4 + b^4 + c^4 - 2b^2c^2 - 2a^2b^2 - 2a^2c^2$ for sides of a triangle

I just got a long expression: $$a^4 + b^4 + c^4 - 2b^2c^2 - 2a^2b^2 - 2a^2c^2$$ and I need to prove its less than zero for every $a$, $b$, and $c$ which are triangle sides I really need tips how to ...
2
votes
2answers
45 views

Equal perimeters of squares and right angled isosceles triangles

Consider a square ABCD having length l and breadth. Now start folding the sides AB and AC so that the figure becomes something like this $$$$ All the vertical and horizontal folds/stairs are equal in ...
1
vote
0answers
45 views

Rationality in Triangle

How can I justify this answer? I think the answer is infinite, but cannot justify it///
2
votes
1answer
33 views

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 ...
2
votes
1answer
48 views

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 ...
0
votes
2answers
41 views

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 ...
5
votes
2answers
186 views

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 ...
1
vote
3answers
37 views

Is this triangle question missing information?

In the $\Delta KLP$, find $a+b$: My question is that: isn't some information missing from the question? Because all I can see is is that $ \usepackage{ gensymb } \angle SKP = \angle LTS = ...
2
votes
2answers
41 views

Sum of areas are equal

Given an equilateral triangle $(ABC)$ and let $P$ be an arbitrary point inside this triangle. Moreover let $V,W,T$ be the orthogonal projections of the point $P$ on to the sides $(AB), (BC), (CA)$ ...
0
votes
1answer
40 views

Length of sides and type of triangle [on hold]

If I have the length of three sides, how do I figure out if it's a right triangle? So what is the formula that will help me find this out?
0
votes
2answers
30 views

Drawing a triangle with 2 known corners and all side lengths

Assume that there are three points $A$, $B$ and $C$. All the pairwise distances are known $(|AB|, |AC|, |BC|)$. But none of the coordinates are known. I want to draw a triangle using those points. ...
0
votes
2answers
41 views

Geometry basic problem

Hy! If i have a triangle with given: b-c=3 cm, a=6 cm and alpha is 30°, how do I draw this? Please help me by telling me where I can find this type of exercises online with explanations. Thank you!
1
vote
1answer
40 views

How to prove these triangle relations?

$O$ is the circumcenter of triangle $ABC$, whereas $G$ is the centroid and $H$ is the orthocenter. $R$ denotes the circumradius. How can I prove the following relations: $OH^2=9R^2-(a^2+b^2+c^2)$. ...
0
votes
0answers
36 views

Moving up the Y axis the lengh of the hypotenuse of a right triangle

If i have a right triangle ABC with B being the right triangle and length AB = 50 and length BC = 50. Based on the Cartesian coordinate system if i wanted to move up the Y axis the length of the ...
3
votes
1answer
38 views

Question about Pasch's Postulate, line going through all three sides of a triangle

I've been reading the textbook Elementary Geometry from an Advanced Standpoint by Edwin E. Moise (3rd ed.). My problem with his wording of Pasch's Postulate, and then a subsequent problem which ...
1
vote
1answer
52 views

How to solve this geometry question?

Let ABC be an acute-angled triangle; L, M, N be the feet of perpendiculars respectively from A, B, C to the opposite sides; D, E, F be the midpoints of the sides BC, CA, AB respectively; and $I_1, ...
1
vote
0answers
30 views

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 ...
1
vote
4answers
71 views

A triangle has to find its third side.

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 $80$. ...
1
vote
1answer
37 views

Crazy rectangles, semi-circles, and circles!

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 ...