1
vote
1answer
17 views

Area of a triangle.

The area of a triangle $ABC$ is $144$.Denote the midpoint of $BC$ by $P$,of $AP$ by $Q$ and of $AC$ by $R$.Calculate the area of the triangle $PQR$. I draw the picture but I do not have any idea to ...
4
votes
2answers
64 views

Is HHH a congurence criteria for triangles?

I wanted to know if a triangle defined by its 3 heights is unique. I took this up as a challenge but was able to get nowhere, can anyone help me? :)
2
votes
1answer
28 views

When is $3R\le 2h_{\max}$ true for acute triangles?

I was working on a problem recently, and it happened that it could be solved if $3R\le 2h_{\max}$ was true for all acute angled triangles. So I used GeoGebra to check it, and found that for some ...
3
votes
3answers
132 views

Drawing a Right Triangle With Legs Not Parallel to x/y Axes?

I have been presented with an interesting problem. How can I decide whether a right triangle with given side lengths can be placed (with integer coordinate vertices) on a Cartesian plane so that the ...
1
vote
1answer
28 views

Find angle and hypotenuse of right angled triangle

Find the missing side and the hypotenuse of a right triangle that has a side length of 5 cm and a perimeter of 30 cm. I'm confused. Can somebody please explain to me how to do this step by step? Not ...
0
votes
2answers
20 views

Configuration of five or more mutually equidistant points in space.

How is it proved that there is no configuration of five or more mutually equidistant points in $R^3$? Is it done by induction? I'm stuck. Help would be appreciated. Well, surely equilateral ...
3
votes
2answers
29 views

Triangle similarity question

I've been trying to solve this question for like 40 mins straight and can't seem to get anywhere. I tried drawing a parallel to |KM| from C to |AB| but that didn't seem to help. I just can't see a ...
0
votes
0answers
24 views

How is the Uniqueness of Equilateral Tetrahedra Proved? [duplicate]

Equilateral tetrahedrons all have this property: For any two of its vertices exists a third vertex, which forms an equilateral triangle with these 2 vertices. (It doesn't necessarily have to be a ...
2
votes
0answers
56 views

Beautiful problem about polyhedrons [duplicate]

A regular tetrahedron has this property: For any two of its vertices exists a third vertex, which forms a regular triangle with these 2 vertices. (But it doesn't mean any 3 vertices form a regular ...
1
vote
0answers
48 views

How to easily prove Euler's theorem, $OI^2=R(R-2r)$?

If $R$ is the circumradius and $r$ is the inradius of some triangle $ABC$, with its circumcenter being $O$ and incenter being $I$, then how to prove: $$OI^2=R(R-2r)$$ I have seen many mentions of ...
-2
votes
0answers
25 views

How many incongruent isosceles triangles can be found in a grid of certain dimensions? [closed]

How many incongruent isosceles triangles can be found in a grid of dimensions $x\times y$? I thought I had the formula, $$\frac{x-1}{2} (y-1)$$ but this does not work when both $x$ and $y$ are ...
0
votes
1answer
23 views

Demonstrate equality: ON = 2m/m-3 in math exercise

I'm actually getting stuck with a part of a quite tricky math exercise using Thales theorem (I've got difficulties with Thales theorem). In this exercise,you have a right handed Cartesian coordinate ...
0
votes
1answer
33 views

Point P on side BC of triangle ABC such that PC=2BP. Find ACB if ABC=45º, APC=60º [closed]

Point P on side BC of triangle ABC such that PC=2BP. Find ACB if ABC=45º, APC=60º. I can't solve this one. Tried some stuff but can't work it out. Can this be done using just simple geometry (like ...
5
votes
1answer
129 views

Symmetrical of a triangle's vertexes

I have the following problem : Show that the symmetrical (ie reflection) of a triangle's vertexes by the opposite side are aligned iff the distance between the orthocenter and the circumcenter is ...
0
votes
1answer
27 views

Side Lengths of Triangles

Die This is Exercise 3-5 from the Art of Problem Solving Volume 2 by Richard Rusczyk and Sandor Lehoczky. I looked at the solution in the solution manual, but I don't quite understand it, so I'm ...
1
vote
1answer
34 views

$H$ is an orthocenter of triangle $ABC$

$H$ is an orthocenter of angle $ABC$. Angle $B$ is $60^{\circ}$. Perpendicular bisectors of $AH$ and $CH$ cross line $AC$ at points $A_{1}$ and $C_{1}$. Show that the centre of $A_{1}HC_{1}$'s ...
1
vote
2answers
46 views

Area of one of four regions within a rectangle

There is a figure below (a rectangle). You can see different colors depicting different regions of the figure. The labels on the top of a region defines the area of that region. Can you find the ...
3
votes
1answer
62 views

Ortocenter and incenter

In triangle $ABC$: $H_{1}$ is a foot of an altitude from side $BC$, $H_{2}$ is a foot of an altitude from side $AC$, $H_{3}$ is a foot of an altitude from side $AB$, $M_{1}$ is midpoint of $BC$, ...
0
votes
1answer
41 views

Dividing a triangle into seventeen equal parts.

I was trying to solve a problem on Pigeonhole principle from Problem Solving Strategies by Arthur Engel. A target has the form of an equilateral triangle with side 2 units. If it is hit ...
0
votes
1answer
26 views

How to prove two triangles have the same centroid?

Suppose you have a triangle ABC and three similar exterior triangles BCX, CAY and ABZ. How can I prove that the centroids of ABC and XYZ are the same point?
1
vote
2answers
73 views

What are the ranges of triangle angles?

Lets say, that $\alpha \le \beta \le \gamma$. As shown here, $60 \le \gamma \lt 180$. What are the minimum and maximum values of $\alpha$ and $\beta$? The answer: $$0\lt \alpha \le 60 \\ 0 \lt ...
1
vote
1answer
33 views

What is the range of angle in front of longest triangle edge?

What is the minimum and the maximum values of the angle $\gamma$ in front of the longest triangle edge?
0
votes
2answers
29 views

Calculate the angle from the given points coordinates.

I'm trying to figure out the way to calculate the a angle value from given coordinates of three points as showed on the illustration below: I know how to ...
0
votes
0answers
40 views

Geometry Problem relating similarity.

Given a triangle $ABC$ and $D$ be a point on side $AC$ such that $AB=DC$, angle $BAC=60-2x$, angle $DBC=5x$ and angle $BCA=3x$ prove that $x=10$. Source: 150 Nice Geometry Problems - Amir ...
0
votes
1answer
20 views

Radius of circumscribed circle of triangle as function of the sides

Given the length ot the sides $a , b$ and $c$ of $ \triangle ABC$. What is the length of the radius of the circumcribed circle? After some formula substitution I came to the monster formula: $$ ...
1
vote
3answers
44 views

If $R$ is the circumradius of $\triangle ABC$, and $\cos A=\frac1{2R}$, $\cos B=\frac1{R}$ and $\cos C= \frac3{2R}$, then is it unique and its area?

Given that $R$ is the circumradius of $\triangle ABC$, and $\cos A=\frac1{2R}$, $\cos B=\frac1{R}$ and $\cos C= \frac3{2R}$. Then would the $\triangle ABC$ be unique? If so how easily we may find its ...
0
votes
3answers
38 views

If the hypotenuse is 4 times the height from A, prove that one of the angles is 15 degrees

In a right triangle (with angle CAB = 90°), suppose |BC| = 4|AD| with AD being the height from A to BC. Prove that the angle BCA is 15°. I had a similar problem but with 22.5°. I thought it would be ...
1
vote
1answer
46 views

Given the coordinates of four points on a plane, how can one determine the shape they form?

Actually it is an algorithm problem, however I cannot solve the problem. So, We have 4 points, how can we know what kind of shape(figure) can be drawn ? I want to learn mathematically. If possible i ...
-1
votes
2answers
45 views

Circle in Triangle Help! [closed]

P is the center of the inscribed circle of right triangle ABC. If AB=4 and AC=2 , find AP.
1
vote
1answer
59 views

Area of similar triangle

Suppose that we are given a triangle whose area is known. put a circle C of radius r inside that triangle. How can we find the area of a triangle similar to the first one and whose inscribed circle is ...
-2
votes
1answer
39 views

Triangle Geometry Help! [closed]

A square is inscribed in a triangle with $\angle A=30^\circ$. What is the ratio of the area of $\triangle EFC$ to the area of $\triangle ADE$?
2
votes
3answers
65 views

How to determine if a triangle can be drawn with the given points.

Given $3$ points $$(x_1, y_1), (x_2, y_2), (x_3, y_3),$$ how does one determine whether they are vertices of a triangle? Thanks.
1
vote
1answer
46 views

Constructing triangle using side length-median relationship

$$\begin{align} m^2_a&=\frac{2b^2+2c^2−a^2}4\\[4pt] m^2_b&=\frac{2c^2+2a^2−b^2}4\\[4pt] m^2_c&=\frac{2a^2+2b^2−c^2}4 \end{align}$$ Solving for $a$, $b$, $c$ in terms of $m^2_a$, $m^2_b$, ...
1
vote
1answer
39 views

Relationship between the altitude of an isosceles triangle and segments drawn to the lateral side from a point on the base.

Question :In an isosceles triangle, the sum of the distances from each point of the base to the lateral sides is constant. I've tried a couple of things, but it seems like this statement is not true. ...
1
vote
2answers
30 views

Relation of length of a projection of a point to a line

In the given figure, can it be said that $x \leq a + b - d$?
0
votes
1answer
16 views

Determine sides of obtuse triangle

I really cannot figure this question out. Can anyone give me a hint please!? Find an integer $a$, for which $a$, $a+1$ and $a+2$ are the lengths of the sides of an obtuse triangle.
0
votes
1answer
59 views

Disprove the possibility of such a triangle.

The image is not that good, but, consider the following figure to be true without actually constructing it,how can one person find a $fault$ in it. The blue colour represents perpendicular, The ...
0
votes
1answer
90 views

How to find coordinates of 3rd vertex of a right angles triangle when everything else is known?

I want to locate precisely the 3rd coordinate of a right angled triangle. I have: the length of three sides The three angles The other two coordinates of the triangle The triangle can lie in any ...
2
votes
0answers
22 views

Self-tiling tile-sets of order 2

A rep-tile is a geometric shape that can be partitioned to smaller copies of itself. The order of a rep-tile is the number of small copies. E.g., a square is a rep-tile of order 4. The smallest ...
0
votes
1answer
24 views

Calculation of third vertices of a triangle given a vector that should be perpendicular to the triangle plane

I have an isosceles triangle (2 sides same length) with vertices O, A and B. OA and OB are the same length. Vertices O and A are known, with O at origin (0,0,0). A known vector V, should be ...
-1
votes
0answers
32 views

Infinite Sum of Areas of Isosceles Triangles Help

Do the areas of the isosceles triangles form a geometric series? I can't figure it out! Triangle $\Delta ABC$ is formed such that $\overline{AC}$ is on the x-axis, $\overline{AB}$ is on the line ...
0
votes
1answer
28 views

If a triangle has side lenghts $a,b,c$ where $c$ is the largest prove that its obtuse if $c^2>a^2+b^2$ and acute if $c^2<a^2+b^2$.

I was thinking about this and I cant get to a formal proof. I have a sort of mental image where you draw $a$ and $b$ perpendicular and the $c$ is too small to connect the two endpoints. So the right ...
0
votes
1answer
22 views

The amount of unit squares being covered

$L$ and $i$ are integers, $L$ is the length of edge of outermost square and $i$ is the minimum length divided from $L$. And there are cells or unit squares consisting the whole block. There is a ...
5
votes
2answers
197 views

What is the flaw in this proof that all triangles are isosceles?

What is the flaw in this "proof" that all triangles are isosceles? From the linked page: One well-known illustration of the logical fallacies to which Euclid's methods are vulnerable (or at least ...
1
vote
2answers
53 views

A simple geometry question

Suppose $ABC$ is any triangle and $BE$ is any line from the vertex $B$ to a point $E$ lying inside the segment $AC$. Let $D$ be any point on $BE$. I would like to verify the following: regardless of ...
0
votes
0answers
23 views

Create dynamic cities of perspective angle x

I'm creating a tilemap... I found you can create unique building sizes with perspective with six tiles using parallel projection, whose angles are always 45 degrees... this allows you to connect to ...
3
votes
3answers
61 views

Right-Angled Isosceles Triangle covering puzzle

Consider a RAIT (right-angled isosceles triangle), from which we remove a RAIT smaller than half its area by a cut perpendicular to the hypotenuse, like this: How many RAITs are required to cover ...
16
votes
7answers
2k views

Can one deduce whether a given quantity is possible as the area of a triangle when supplied with the length of two of its sides?

Recently I have found a question like following: In triangle $ABC$, $AB=AC=2$. Which of the following could be the area of triangle $ABC$? Indicate all possible areas: [A] $0.5$ [B] $1.0$ ...
1
vote
2answers
50 views

Prove triangles formed by two midpoints and an altitude are congruent

Triangle ABC has altitude BH. M is the midpoint of AB, and N is the midpoint of CB. Prove triangle MBN is congruent to triangle MHN. Can we say that MN bisects BH? If so, why? If MN bisects BH (at ...
1
vote
1answer
50 views

Prove triangle made from two altitudes and midpoint is isosceles

In triangle ABC, AH and BK are altitudes. M is the midpoint of AB. Prove that triangle MHK is isosceles. All I can see is that the angles formed where the altitudes intersect are equal, and since ...