For questions about properties and applications of triangles

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1answer
34 views

Is there enough information to answer this question?

My daughter got this question and I cannot solve it - or even give her direction. It appears there in not enough information. the number of equilateral triangles of side 1 into which an equilateral ...
-1
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2answers
7 views

Angles and polygons.

The ratio of the interior angle to the exyerior angle of a regular polygon is 5:2. Find the number of sides of the polygon. Note the polygon may be three sided
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1answer
20 views

Relations involving the altitudes and orthocenter of a triangle [on hold]

For acute $\triangle ABC$ with altitudes $AD$, $BE$, $CF$, orthocenter $H$, and area $S$, I have to prove that: $$AB^2 + HC^2 = BC^2 + HA^2 = AC^2 + HB^2 \tag{a}$$ $$AB \cdot HC + BC \cdot HA + ...
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1answer
34 views

Maximum number of equilateral triangles in a circle

I am stuck with a question. Given a circle with radius $x$ cm, what is the maximum number of equilateral triangles of side length 1 cm that can fit in the circle without overlapping or ...
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2answers
23 views

Right triangle trigonometry help?

I've got a right triangle where I know the slope of side $c$ based on the two points $(-150,200)$ and $(0,0)$. Also I know the length of side $a$. I was wondering based on these two known factors how ...
0
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1answer
25 views

Dividing a Triangle by Connecting the Midpoints of its Sides

If $T$ is any triangle. Suppose we connect the midpoints of its sides forming four triangles. Does these four triangles have the same angles?
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1answer
30 views

If $AD=999$ and $PQ=200$, find the sum of the radii of those incircles.

Let $ABCD$ be a convex quadrilateral with $\angle DAB =\angle BDC=90° $. Let the incircles of $\Delta ABD $ and $ \Delta BCD $ touch $BD$ at $P$ and $Q$, respectively with $P$ between $Q$ and $B$. If ...
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2answers
14 views

Finding vector form of an angle bisector in a triangle

Find vector form of angle bisector, $\vec{BP}$, using $\vec{b}$ and $\vec{c}$. That's how far I've got. Please don't use $tb+ (1-t)b$, or similar since I don't know what that is. Just basic dot ...
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0answers
34 views

Definition of triangle

If a polygon has 3 sides, but one side has zero length (or one angle is zero degree), is it still a triangle by definition of triangle? and how about if it has 2 sides, if not 3 sides are zero?
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0answers
45 views

“Natural” labeling of triangles

The angles of a triangle are (capital) $A,B,C$ and the lengths of the sides are (lower-case) $a,b,c$. At your mother's knee, you were taught that the side whose length is called (lower-case) $a$ ...
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1answer
33 views

Two cevians divide a triangle into 4 parts. Calculate the area of the 4th part, given the other 3.

Good day Here is the question: Connecting $AF$ and setting areas $\triangle ADF = x$ and $\triangle AFE = y$: $\frac {9+x}{12} =\frac y{15}$ $\frac{15+y}{12} =\frac x9$ from the ratios of the ...
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0answers
23 views

Line tangent to circle inside an isosceles triangle

If you take a circle enclosed inside an isosceles triangle, and then draw a line which is tangent to the circle and which intersects with the two equal sides, is that line parallel to the triangle's ...
2
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1answer
56 views

$ABCD$ is a square. $M$ and $N$ are points on $AB $ and $BC$, respectively such that $\angle MDN=45^\circ$…

$ABCD$ is a square. $M$ and $N$ are points on $AB $ and $BC$, respectively such that $\angle MDN=45^\circ$. $R$ is the midpoint of $MN$ and $P$ and $Q$ are the points where $AC$ is intersected by $DM$ ...
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2answers
27 views

how to find the number of integer coordinates in the interior of triangle

How to find the number of integer coordinates in the interior of the triangle with vertices(0,0) (0,21) (21,0).
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2answers
45 views

A Part of a semicircle between the two legs of a right angle triangle

In a right angled triangle, a semicircle is drawn such that its diameter lies on the hypotenuse and its center divides the hypotenuse into two segments of lengths 15 and 20.Find the length of the arc ...
-1
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0answers
31 views

Triangle inequality $ax ≥ br + cq$

I got stuck on this problem : Given a triangle (△ABC) of sides $a$, $b$ and $c$, let $O$ be a point inside △ABC. Let $D$, $E$ and $F$ be points on sides a, b and respectively c such that $OE ⊥ ...
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2answers
68 views

Find x in below diagram geometry

I am having difficulty in solving below question. Please help. Find x angle in below diagram I have drawn two parallel lines from D and E intersecting sides CB and CE respectively on F and G. look ...
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2answers
49 views

Angle chase:In $\Delta ABC, AB=AC $ and $\angle BAC=20°.$ If $CD$ is the median from $C$ to side $AB$, find $\angle ADC$.

In $\triangle ABC, AB=AC $ and $\angle BAC=20^\circ$ If $CD$ is the median from $C$ to side $AB$, find $\angle ADC$.
3
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0answers
37 views

Area of $A'B'C'$ is to area of $ABC$ is $\frac{(m-n)^2}{m^2+mn+n^2}$

In the sides $BC,CA,AB$ are taken three points $A',B',C'$ such that $BA':A'C=CB':B'A=AC':C'B=m:n$.Prove that if $AA',BB',CC'$ are joined they will form by their intersections a triangle whose area is ...
3
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1answer
41 views

Prove that the ratio of the areas of the triangles $A'B'C'$ and $ABC$ is $2\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$

If the bisectors of the angles of a triangle $ABC$ meet the opposite sides in $A',B',C'$,prove that the ratio of the areas of the triangles $A'B'C'$ and $ABC$ is $2\sin \frac{A}{2}\sin \frac{B}{2}\sin ...
2
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3answers
40 views

Triangle area inequalities

I've got stuck on this problem : Proof that for every triangle of sides $a$, $b$ and $c$ and area $S$, the following inequalities are true : $4S \le a^2 + b^2$ $4S \le b^2 + c^2$ ...
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1answer
41 views

Don't fence me in [closed]

A farmer owns a piece of land in the shape of an equilateral triangle, 200 m on a side, which is totally fenced in. He wishes to construct an additional fence parallel to the side fronting the road so ...
2
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4answers
45 views

Inequality between altitude and sides in triangle

Let $a,b,c$ be the side lengths and $h_a,h_b,h_c$ the altitudes each connect a vertex to the opposite side and are perpendicular to that side. Then we need to prove ...
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0answers
20 views

Proof of Menelaus using areas

I've tried to proof Menelaus' theorem using areas, but I've didn't figure out how. Some suggestions would be appreciated. Menelaus' Theorem states : Given a triangle ABC and a transversal ...
2
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3answers
40 views

Proving a ratio that has a relation with the Perpendicular bisectors and circumcircle

$ABC$ is a triangle, $D$ is a point on the side $BC$ of $\triangle ABC$, $R_b$ is circumradius of $\triangle ABD$ , and $R_c$ is the circumradius of $\triangle ACD$. Prove that $$ {Rb\over Rc} ...
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1answer
38 views

Prove by vector method that $p_1+p_2=p_3$

Let $ABC$ be an acute angled triangle whose incenter and centroid are respectively $I$ and $G$.$AI,BI$ and $CI$ cuts the sides of the triangle at $P,Q,R$ respectively.If $p_1,p_2$ and $p_3$ are the ...
0
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1answer
25 views

Using Tan to find the area of a triangle

I have come across a question that I can't seem to figure out. If tanA = 3/4, find the area of the given triangle without using a calculator The given triangle is an scalene triangle with a ...
4
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3answers
403 views

Find the longest side of the triangle.

The sides $a,b,c$ of a $\triangle ABC$ are in $GP$ whose common ratio is $\frac{2}{3}$ and the circumradius of the triangle is $6\sqrt{\frac{7}{209}}$.Find the longest side of the triangle. I used ...
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1answer
79 views

Geometry - angle bisector, circumcircle: SL olympiad

I tried this problem as much as I can, but I got nothing. This is a Sri Lankan mathematical olympiad problem. Let $P$,$Q$ be points on the sides $AB$ and $AC$, respectively, of a $\triangle ABC$ ...
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1answer
236 views

The case of Captain America's shield: a variation of Alhazen's Billard problem

I'm sure a lot of you are acquainted with Alhazen's Billiard problem, which involves finding the point on the edge of a circular billiard table at which a cue ball at a given point must be aimed in ...
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5answers
426 views

What is the least number of (fixed) parameters I can ask for, when calculating area of a triangle of unknown type?

I need to calculate the area of a triangle, but I don't know, whether it is right angled, isoscele or equilateral. What parameters do I need to calculate the area of a triangle of unknown type?
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2answers
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How to scale a triangle such that the distance between original edges and new edges are equal?

This is very similar to this question: Coordinates of parallel triangle with a distance of 'd' between the parallel edges? That seems to provide the answer in 2D, but I am unsure how to ...
0
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1answer
36 views

Find triangle $ABC$ satisfies $1+2\sqrt{2}\sin\frac{B}{2}\sin\frac{C}{2}=\cos B+\cos C$

$\color{Red}{\texttt{Find all the triangle ABC}}$ whose angles satisfies $$2\left (1+tan^2\frac{C}{2} \right )\left [ cos^2\left (\frac{13\pi }{2}+\frac{B}{2} \right ...
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2answers
32 views

Find area of an isosceles triangle with medians perpendicular to each other and base of length 4. [closed]

Hello guys can you help me with this problem.thanks in advance.
6
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3answers
167 views

Prove that $\angle AQP+\angle NAP=90^o$

Let a triangle $ABC$ be right at $A$, $AH$ be the altitude to the side $BC$. Let $M$ be an arbitrary point located in $AH$. Draw circle $B$ with the radius $BA$, circle $C$ with the radius $CA$. ...
2
votes
2answers
61 views

Triangle Area problem

I've been trying to solve the following: Let $ABC$ be a triangle with sides $a, b $ and $ c$, inradius $r$ and exradii $r_a, r_b$ and $r_c$. If $A'B'C'$ is another triangle with sides $\sqrt{a}, ...
0
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1answer
46 views

$\frac{1}{\Delta}\sqrt{abc(a+b+c)}$

If the triangle ABC has sides $a,b,c$ opposite to the vertices A,B,C respectively and $\Delta$ is the area.The expression $\frac{1}{\Delta}\sqrt{abc(a+b+c)}$ is $(A)\leq16\hspace{1 ...
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0answers
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. Find the projection of the triangle on the coordinate planes.

Given the following, three vectors: a⃗ =3i−2j+5k b⃗ =i−6j+6k c⃗ =2i+3j−k Relative to cartesian coordinate systems with origin O. I calculated the sides to be 4.58,11.45 and 7.87. I also calculated ...
2
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1answer
49 views

Find all integer sided triangles whose area equals their perimeter.

Find all integer sided triangles whose area equals their perimeter? Taking sides to be $e,f,g$ we have, $$\sqrt{(e+f+g)(e+f-g)(e+g-f)(g+f-e)}=4(e+f+g)$$ how to proceed next ?
6
votes
3answers
174 views

Let $M$ be an arbitrary point located inside the triangle $ABC$. Prove that $\cot\angle MAB + \cot\angle MBC + \cot\angle MCA \geq 3\sqrt{3}$

Let $M$ be an arbitrary point located inside the triangle $ABC$. Prove that $$\cot\measuredangle MAB + \cot\measuredangle MBC + \cot\measuredangle MCA \geq 3\sqrt{3}$$
2
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1answer
54 views

IO is inclined to BC at an angle $\tan^{-1}(\frac{\cos B+\cos C-1}{\sin B-\sin C})$

Show that the line joining the inscribed center to the circumscribed center of a $\triangle ABC$ is inclined to BC at an angle $\tan^{-1}(\frac{\cos B+\cos C-1}{\sin B-\sin C})$ I tried taking the ...
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2answers
55 views

Line joining the orthocenter to the circumcenter of a triangle ABC is inclined to BC at an angle $\tan^{-1}(\frac{3-\tan B\tan C}{\tan B-\tan C})$

Show that the line joining the orthocenter to the circumscribed center of a triangle ABC is inclined to BC at an angle $\tan^{-1}\left(\frac{3-\tan B\tan C}{\tan B-\tan C}\right)$ I let the foot of ...
0
votes
1answer
40 views

How do you determine the number of triangles using SSA?

Suppose I'm given 2 sides and an angle for a triangle. How do I use those sides to determine whether the measurements can give 0, 1, or 2 triangles? Do I use Law of Sines or Cosines?
1
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1answer
26 views

$\frac{CE}{DE}=\frac{(a+b)^2}{kc^2}$

If bisector of angle C of an acute triangle ABC cuts the side AB in D and the circumcircle of triangle ABC in E,then $\frac{CE}{DE}=\frac{(a+b)^2}{kc^2}$.Then what is value of k? Since angle bisector ...
0
votes
1answer
47 views

Minimizing the area of the triangles containing a square of side $1$

This exercise is from a past admission exam to an Italian institute: Among all the triangles that contain a square of side $1$, which ones have minimum area? I have solved it, however I'd like ...
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0answers
27 views

Prove that $r^3\rho=2R \rho \rho_1\rho_2 \rho_3$

I doubt whether this question is correct or not. Because in the LHS, it is fourth dimension in length and in the RHS, it is fifth dimension in length. If correct, I don't know how to prove it. If ...
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4answers
64 views

Find all triangles with a fixed base and opposite angle

I have a situation where I know the cartesian coordinates of the 2 vertices of a triangle that form its base, hence I know the length of the base and this is fixed. I also know the angle opposite the ...
2
votes
1answer
70 views
1
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1answer
26 views

Prove that in every acute triangle, the equation stands: $h_c = c \cfrac{\tan(\alpha)\cdot \tan(\beta)}{\tan(\alpha)+\tan(\beta)} $

Prove that in every acute triangle, the equation stands: $$h_c = c \cfrac{\tan(\alpha)\cdot \tan(\beta)}{\tan(\alpha)+\tan(\beta)} $$
7
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2answers
163 views

Find max of $S = x\sin^2\angle A + y\sin^2\angle B + z\sin^2\angle C$

Let $x$, $y$, $z$ are positive constants. $A$, $B$, $C$ are three angles of the triangle. Prove that $$S = x \sin^2 A + y \sin^2 B + z \sin^2 C \leq \dfrac{\left(yz+zx+xy\right)^2}{4xyz}$$ and find ...