# 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 .
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### 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_______.
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### 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 $\triangle ABC$.
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### 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 ...
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### 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 ...
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### 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 ...
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1 vote
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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1 vote
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
1 vote
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### 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, ...
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### 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 ...
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### 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 ...
1 vote
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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