Questions tagged [triangles]
For questions about properties and applications of triangles.
7,205 questions
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Find a synthetic proof to an old problem .
I found this problem in a French paper translated from Arabic in 1927 by an author named Al Bayrouni . I wonder if it can be found in one of Archimedes' works . Here is the statement :
ABC is a ...
- 1,816
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Basic Proportionality Theorem/ Thales Theorem [closed]
The Basic Proportionality Theorem seems so obvious but the construction to prove it (drooping perpendicular to equate areas) is not at all obvious to me. Can anyone tell how to prove this Theorem in a ...
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Prove that $BN=LC$. A geometry problem from the national round of math olympiad.
Problem: Let $ABC$ be an acute triangle and $D$ be the foot of the altitude from $A$ onto $BC$. A semicircle with diameter $BC$ intersects segments $AB, AC$ and $AD$ in the points $F, E$ and $X$, ...
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Find the ratio $\frac{AC}{BC}$ given a specific configuration of equilateral triangles around a right triangle (need Euclidean geometry approach)
I encountered a geometry problem involving a right-angled triangle and several constructed equilateral triangles. I am trying to solve the second part of the problem (Case 2 in the image).
Continues ...
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Maximizing the area of a triangle
I am given 3 radii $r_a, r_b, r_c$ and I want to determine the 3 angles $\phi_a,\phi_b,\phi_c$ for which the area of the triangle defined by $\left(r_a\cos(\phi_a),r_a\sin(\phi_a)\right),\,\left(r_b\...
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Find the ratio of side lengths of two equilateral triangles given a midpoint condition
Problem Statement:
As shown in the diagram below, we have two equilateral triangles, $\triangle ABC$ and $\triangle ADE$, sharing a common vertex $A$.
We construct a line connecting vertices $B$ and $...
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Vieneuous Triangle Theorem: Point on median with half-altitude distance sum in isosceles triangle [closed]
Proposing the Vieneuous Triangle Theorem (Nov 24, 2025, from Vietnam).
Theorem: In isosceles $\triangle ABC$ ($AB=AC$), median $AD$ is from $A$ to base midpoint $D$. There exists unique $P$ on $AD$ s....
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What is the length of the height AH?
I'm trying to find a problem about right triangles with a minimalist statement that isn't too obvious. Here's what I've come up with :
ABC is an A–right triangle, H is the orthogonal projection of A ...
- 1,816
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Similarity argument in right triangle with perpendiculars to sides
I have a right triangle $OAB$ with right angle at $O$, and let
\begin{equation}
OA = L, \quad OB = 1.
\end{equation}
Let $a$ be a point on $OA$ and $b$ a point on $OB$. From these points, ...
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Find $\angle C$ given the relation $a^4 + b^4 + c^4 = 2a^2c^2+2b^2c^2$ [closed]
In a $\triangle ABC$ with sides $a, b, c$, the following relationship holds:
$$a^4 + b^4 + c^4 = 2a^2c^2+2b^2c^2$$
I need to determine the possible values for angle $C$.
My Attempt:
I suspect this ...
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Find the length of the side DQ
In rectangle $ABCD$, points $P$ and $Q$ lie on sides $AD$ and $DC$ respectively, such that $AP = 2 \times DQ$.
Given that $AB = 5\,\text{cm}$, $BC = 10\,\text{cm}$, and the area of quadrilateral $BPQC$...
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Can $1, \sqrt{26},5$ construct a right angle triangle?
Problem: Which $3$ sides do not create a right angled triangle?
A. $12,13,5$
B. $5,3,4$
C.$1, \sqrt{26},5$
D. $6,8,10$
Context: It was a mcq from my math test. But, I was confused to see that all ...
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Can an isosceles triangle with a $60^\circ$ angle be proven equilateral independently of the triangle angle sum theorem?
There is a famous theorem in elementary geometry:
Theorem. An isosceles triangle with a $60^\circ$ angle is equilateral.
Two cases of this theorem are depicted below. I consider any (or both) of ...
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Is a collinearity step missing in this Miquel point proof?
Problem:
Solution:
Question: The problem and solution are taken from the book A beautiful journey through olympiad geometry. The problem is from the chpater $19$, complete quadrilateral. In the ...
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Finding the angles of a non-equilateral $\triangle ABC$ with centroid $G$ such that $\angle GAB=\angle GCA=30^\circ$
In the attached figures, $G$ is the centroid of $\triangle ABC$.
When this triangle is equilateral ( fig 1) , it is obvious that each of the two angles shown in this figure measures $30^\circ$.
Using ...
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