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Questions tagged [triangles]

For questions about properties and applications of triangles.

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How do I determine angles and lengths of a triangle if I'm only given one angle and one side length. [closed]

I just don't think I can if given angles A, B, and C (where A and B are unknown and C = 90 deg) and sides a, b, and c--the sides opposite the angles--where side a is 10 in and sides b and c are ...
Rex Miller's user avatar
-2 votes
0 answers
47 views

Olympiad geometry problem with angles [closed]

Triangle ABC has a right angle at A. Altitude AD has length 20. The bisector of angle B meets AD at K. If angle ACK= 2angle DCK, find KC. This is an olympiad practice problem which I am trying to ...
Alex Yao's user avatar
-2 votes
0 answers
36 views

How can I solve this geometry problem? [closed]

Triangle ABC is equilateral triangle. M is on side AB and P is on side CB such that MP || AC. D is the centroid of triangle MBP and E is the midpoint of PA. Find the angles of triangle DEC.
Sohan Sen's user avatar
4 votes
1 answer
59 views

In a geodesic triangle, is the longest side opposite to the largest angle?

If I have a complete (smooth) Riemannian manifold $(M,g)$ and three points on it, that I connect with distance minimizing geodesics, will the longest edge be opposite to the largest angle? In ...
PleaseAnswerMyQuestion's user avatar
-5 votes
0 answers
22 views

What are sides of triangle if angular trisector trisects opposite side in particular segments [closed]

In triangle ABC, AD and AE trisect ∠BAC. The lengths of BD,DE and EC are 2,3 , and 6 , respectively. Find the length of the shortest side of △ABC .
biswarup datta pramanik's user avatar
-2 votes
0 answers
27 views

How to find hypotenuse from the distance of the centroid to that point that is the right angle [closed]

I have a problem that says: In Rt△ABC, ∠C= 90°, point G is the centroid of Rt△ABC. If CG=6, then the length of the Hypotenuse is_______.
Abel Ma's user avatar
-2 votes
0 answers
22 views

Finding the area of a triangle knowing the coordinates of the midpoints of its medians [closed]

The midpoints of the medians of $\triangle ABC$ are $(1,2)$, $(4,4)$, and $(2,8)$. Find the area of the $\triangle ABC$.
Daigo Hideoshi's user avatar
2 votes
4 answers
106 views

Is there is a formula to calculate the coordinates of the orthocenter of a triangle?

I'm trying to find the coordinates of the orthocenter (the intersection point of all altitudes) of a triangle given its vertices' coordinates $A=(x_1, y_1), \ B=(x_2, y_2) , \ C=(x_3, y_3)$. I ...
pie's user avatar
  • 6,546
-1 votes
2 answers
67 views

Proof using Converse of Thales Theorem for isosceles right-angled triangle

Let $ABC$ an isosceles right-angled triangle with the right angle at $C$. Suppose that the points $D$ and $E$ lie outside the triangle on the half-line $AC$ and $CB$, respectively (see picture). Let ...
user267839's user avatar
  • 7,499
0 votes
0 answers
72 views

Maximum area a traingle can have which can fit inside a circle of radius $r$? [duplicate]

So what is the maximum area of a triangle which can fit inside a circle of radius r? My first approach: We know that $\text{ Circumradius }=\frac{abc}{4×\text [area-of- triangle}$ (here abc are side ...
Guess's user avatar
  • 169
1 vote
1 answer
42 views

2-D scalene obtuse triangle trigonometry.

I am struggling with this trigonometry question: I tried using the cosine law with angle DBC $a^2 = b^2 + c^2 - 2bc \cos A$ but you need to know the measure of the angle. In terms of the angle Φ, the ...
Tanish Shukla's user avatar
7 votes
4 answers
200 views

Largest Area Triangle in the Vesica Piscis

I can place any three points in or on a vesica piscis1. I wish to find the triangle of maximum area. I know the area of the vesica piscis is $(\frac{2π}{3}-\frac{\sqrt{3}}{2})d^2$ (where d is the ...
WakkaTrout's user avatar
0 votes
1 answer
46 views

Knowing a side, the inradius, and the circumradius of a triangle, find the other two sides [closed]

I need help with this easy triangle problem: We know: One of the sides a = 16 cm. The inradius r = 6cm. And the circumradius R = 17 cm. That's all. We must find the lengths of the other two sides. ...
Georgi Angelov's user avatar
2 votes
1 answer
72 views

Determine the angle $\angle DEC$ in a triangle (Euclidean Geometry)

Any ideas how to find the angle $\angle DEC$ in the following situation shown in the image: In the above figure we have that $\angle BAC = 90, \angle ABD = \alpha, \angle DBC = 2\alpha$, and $\angle ...
ChrisNick92's user avatar
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1 vote
2 answers
40 views

Alternate Proof for Sum of Sides of a Triangle Inequality

I recently stumbled upon an idea for a proof for the sum of two sides of a triangle inequality. Note that I am just a high school student and feel free to correct me wherever if I am wrong. Statement/...
Rishwanth's user avatar
0 votes
3 answers
61 views

In the convex quadrilateral $ABCD$ Assuming that $\angle BCD< 90^{\circ}$, prove that:$\angle DAB< 90^{\circ}$

In the convex quadrilateral $ABCD$, with its side lengths $AB$, $BC, CD$, are $25, 39, 52$, and $DA$ $60$ units, respectively. Assuming that $\angle BCD< 90^{\circ}$, prove that:$\angle DAB< ...
user62498's user avatar
  • 3,586
1 vote
0 answers
37 views

Proof of Thomson cubic pivotal property without coordinates

The Thomson cubic is defined as the cubic going through A,B,C, the three side midpoints, the three excenters. Is there a way to prove its pivotal property (any two isogonal conjugates on it have a ...
user118161's user avatar
1 vote
3 answers
219 views

Parallel line equation

I want to incorporate 2 diagonal lines in a logo design. The lines have to be parallel to each other and have to be exactly 0.5 inches apart when measured perpendicular. The upper point of Line 1 has ...
Geo's user avatar
  • 37
2 votes
0 answers
33 views

What is the maximum area of n non-overlapping equal area triangles inscribed in a circle of radius

What is the maximum area of n non-overlapping equal area triangles inscribed in a circle of radius 1? For n = 1, the triangle is equilateral. For n = 2, we have 2 isosceles right triangles sharing a ...
Ultima Gaina's user avatar
0 votes
1 answer
38 views

Proving Symmedian intersects intersection of tangents

I'm going through Evan Chen's "Euclidean Geometry in Math Olympiads" and I've come to Chapter 4's section on Symmedians. Proposition 4.24 says: Let $X$ be the intersection of the tangents to ...
PabloGamerX's user avatar
3 votes
2 answers
292 views

What is the minimum value of $a+b-c$ in a triangle with a fixed area?

Let $\Delta$ be the fixed area of a triangle inscribed inside on a fixed circle of radius $R$. The sides of the triangle $(a,b,c)$ are unknown. We want to estimate a the lower bound of the triangle ...
Nilotpal Sinha's user avatar
0 votes
0 answers
36 views

acute angles $\alpha$ and $\beta$ of the triangle $ABC$ satisfy $\sin^2 \alpha + \sin^2 \beta = \sin (\alpha + \beta)$, then $ABC$ is right-angled. [duplicate]

Given that the acute angles $\alpha$ and $\beta$ of the triangle $ABC$ satisfy the condition $\sin^2 \alpha + \sin^2 \beta = \sin (\alpha + \beta)$. Prove that the triangle $ABC$ is right-angled. ...
user avatar
3 votes
3 answers
193 views

Calculate the length of segment $AD$.

Given a triangle $ABC$ and its circumscribed circle, point $E \in BC$. Let $D$ be the intersection of the circle and line $AE$ (see the figure). Also, let $|AB| = |AC| = 12$ and $|AE| = 8$. Calculate ...
user avatar
0 votes
0 answers
47 views

Finding all empty triangles of a plane

I have a set of $N$ points ${(x_i,y_i)}_{i=1,...,N}$. I am looking for an efficient algorithm to find the set of all empty triangles (i.e., that do not contain any points). The brute-force method that ...
Quentin PLOUSSARD's user avatar
1 vote
3 answers
132 views

Find the total area of two triangles within a square

I solved this but others have conflicting answers, I'd love some validation: Total area of the green triangles. Please show your work on how to solve this - lots of Pythagoras and more needed. Ok, ...
Mangobubbly's user avatar
3 votes
2 answers
81 views

calculus optimization problem: rectangle inscribed in a triangle.

I have a solution to the problem below from my course materials, but I cannot understand where I went wrong with my own attempt at a solution. Any advice much appreciated. Problem: Given a right ...
Chris Bedford's user avatar
0 votes
1 answer
77 views

Find the segment BT in the triangle inscribed below

In the figure, $AB.BC = 60$ and $BT.TP = 40$. Calculate BT with B and T tangency points. (Answer:$2\sqrt5$) I try: $AT.TC = BT.TP \implies AT.TC = 40$ $AM.AB = AT.AC$ $AT^2 = AM.AB \implies AT^2 = AT....
peta arantes's user avatar
  • 6,947
2 votes
1 answer
79 views

Isosceles triangle calculations

We have that if $𝑆 ⊂ ℝ^2$ is a set of $𝑛$ points in the plane, with no three points on a common line, then there exists a point $𝑎 ∈ 𝑆$ that determines at least $(𝑛 − 1)/3$ distinct distances to ...
D. S.'s user avatar
  • 148
0 votes
1 answer
35 views

Understanding the geometry behind finding the area of a triangle defined by three vertices in three space

I am self-studying linear algebra using Jim Hefferon's Linear Algebra textbook (published in 2020). As I was making my way through the determinants section, I stumbled upon the following question: ...
LateGameLank's user avatar
-1 votes
1 answer
49 views

Pythagorean triangles with non rectilinear [closed]

Does there exist a Pythagorean triple whose corner is non-rectilinear with axes, yet all three vertices are on integer coordinates? The question was closed due to "lack of context". The ...
Dwayne Towell's user avatar
3 votes
6 answers
230 views

Rotating and scaling an arbitrary triangle such that the new triangle has its vertices on the sides of the original one

Given $\triangle ABC$, and a scale factor $r \lt 1 $, I want to find the necessary rotation (center and angle) such that the rotated/scaled version of the triangle has its vertices lying on the sides ...
Quadrics's user avatar
  • 24.4k
0 votes
2 answers
108 views

Find the missing angle in the triangle below if the sum of two sides is given

I'm having a hard time solving this problem, so if anyone would be kind enough to point me in the right direction, I'd be forever grateful! Find the value of the the missing angle in the triangle ...
PinkBlack's user avatar
0 votes
1 answer
82 views

If A and C are nearer, is B nearer too?

Let $A, B, C$ be three colinear points in $\mathbb{R}^2$, and $P, Q$ any two points in $\mathbb{R}^2$ (on the same line or not). I am trying to prove (or disprove) the following lemma, where $d$ is ...
Erel Segal-Halevi's user avatar
3 votes
1 answer
136 views

An "almost" geodesic dome

A regular $ n$-gon is inscribed in the unit circle centered in $0$. We want to build an "almost" geodesic dome upon it this way: on each side of the $n$-gon we build an equilateral triangle ...
user967210's user avatar
10 votes
2 answers
503 views

A circle is inscribed in a triangle, with three other circles in the corner regions. The radii are integers. Possible values of the largest radius?

Consider four circles with integer radii inscribed in a triangle as shown. That is, a circle with integer radius $R$ is inscribed in a triangle, and three other circles with integer radii $a,b,c$ are ...
Dan's user avatar
  • 25.7k
4 votes
3 answers
186 views

Find the sum $PC^2+PD^2$ in the trapezoid inscribed below

If there is a semicircle of diameter $AB$ in which an isosceles trapezoid $ABCD$, ($AB \parallel CD$) is inscribed. On $AB$, we take a point "$P$" such that $PA^2 + PB^2 = 5^2$. Calculate: $...
peta arantes's user avatar
  • 6,947
0 votes
0 answers
30 views

foot of perpendicular using cosine

We consider a triangle $\Delta_\textbf{ABC}$ in $\mathbb{E}^2$ and use the usual notations for the side lengths and the angles of the triangle. Let $\textbf{L}$ be the foot of the perpendicular from $\...
tom31415's user avatar
10 votes
2 answers
243 views

Proving 2 triangles are congruent

Given $\Delta ABC, \Delta A'B'C'$ s.t $\widehat{BAC}=\widehat{B'A'C'}, BC=B'C', AD=A'D'$ $(AD, A'D'$ are internal bisectors of $\widehat{BAC}, \widehat{B'A'C'}$ respectively). Prove that $\Delta ABC=\...
Harry's user avatar
  • 101
3 votes
5 answers
94 views

Minimizing $\left(\frac{c}{a} + \frac{c}{b}\right)^2$, where $c$ is the hypotenuse of a right triangle with legs $a$ and $b$

This question is regarding the following problem Given that $a, b, c$ are the sides of the $\triangle ABC$ which is right angled at $C$, then what is the minimum value of the following expression? $$\...
koiboi's user avatar
  • 356
2 votes
1 answer
35 views

Given arbitrary number of triangles intersecting pairwise over hexagonal regions prove that intersection of all these triangles has positive area

Given arbitrary number of triangles intersecting pairwise over hexagonal regions prove that intersection of all these triangles has positive area. Hexagonal intersection means that every side of two ...
Vladimir_U's user avatar
3 votes
1 answer
53 views

In a triangle ABC : 2 externaly tangent circles, also tangent to BC with centers on line segments AB and AC : envelope of their lines of centers?

The figure here gives an illustration of the configuration described in the title in 4 cases ; consider especialy the fourth one, materialized by red circles, red center points, and a red line segment ...
Jean Marie's user avatar
  • 83.9k
0 votes
1 answer
108 views

Trigonometry Question: $\sqrt{13-12\cos x}+ \sqrt{7-4\sqrt3\sin x }=2\sqrt3$ [closed]

![Sqrt(13-12cosx) + sqrt(7-4sqrt3sinx =2sqrt3][1] Hi I have started solving the question I have constructed a triangle with base as $3$ and height as $\sqrt3$. Both terms gives a cosine formula then I
Noobplayer's user avatar
0 votes
1 answer
50 views

Proving $pa^2+qb^2+rc^2\ge 4\sqrt{pq+qr+rp} \cdot S_{\Delta ABC}$ [closed]

I want to prove the inequality: $$pa^2+qb^2+rc^2\ge 4\sqrt{pq+qr+rp} \cdot S_{\Delta ABC}$$ $a,b,c$ are the three sides of $\Delta ABC$, $ p,q,r\in R^+$ . I think this is a variant of Weisenböck's ...
Lichium's user avatar
4 votes
3 answers
145 views

Can the center of circumscribed circle in a triangle be on the incircle?

Besides the obvious answer of an isosceles right triangle, can there be other triangles where the center of its circumscribed circle is located on the perimeter of its incircle? I tried using the ...
Sadra Daneshvar's user avatar
1 vote
1 answer
35 views

What is the relationship between the silver ratio and the postion of a circle in the corner of a triangle?

I was recently trying to figure out how much to offset a circle in the corner of a right-angled triangle and found empirically that the x-offset needed to be around 2.414 which I later found to be the ...
TyghtMo's user avatar
  • 13
0 votes
2 answers
114 views

Four Circles radii

Consider following constellation of four adjacent circles. Question:(Initial question doesn't give an unique solution; see edit) Assume we know the radii $R_1,R_2,R_3$. Is there a geometric/synthetic ...
user267839's user avatar
  • 7,499
0 votes
0 answers
68 views

Geometric Inequality with Angle Bisectors in a Triangle

Given triangle ABC with angle ABC = 60°. AP is a bisector of angle BAC. AQ is a bisector of angle CAP. Prove that BC > 4PQ. So far, I've managed to express the equality of the products of sides ...
Tan's user avatar
  • 11
1 vote
2 answers
84 views

A geometry problem involving three altitudes of a triangle.

I think this is a well-known result in plane geometry but I don't remember how to solve it. So I decided to post it here, hoping that I would get a hint or a solution if possible. Let $ABC$ be an ...
anonimo's user avatar
  • 499
1 vote
0 answers
92 views

If any two of the circumcenter, incenter, orthocenter, and centroid are the same point, is the triangle equilateral?

Circumcenter- the intersection of the perpendicular bisectors of a triangle, from which the length to each of the vertices is equal. Incenter- the intersection of the angle bisectors of a triangle, ...
Problem_Solving's user avatar
2 votes
1 answer
55 views

How to inscribe a triangle with side-length 1 into a hyperbola, analogous to inscribing a triangle with hypotenuse 1 into a circle?

I have recently been looking at the Hyperbolic Trigonometric functions, and I've run into this unique dilemma which I haven't seen a satisfying answer to. I would ask that you watch this video before ...
HedgehogusSpeedicus's user avatar

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