Questions tagged [angle]
An object formed by two rays joining at a common point, or a measure of rotation. In the latter form, it is commonly in degrees or radians. Please do not use this tag just because an angle is involved in the question/attempt; use it for questions where the main concern is about angles. This tag can also be used alongside (geometry).
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Difference of length two segments
I am trying to solve this question (AIMO 2024 Grade 8, question 20):
In figure, $D$, $E$ are points on $BC$ and $AC$ respectively. If $\angle BAD=\angle CAD$, $\angle ABE=\angle CBE=\angle ACB$, $AE=...
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Find the other two angles of a triangle when one angle and the ratio of two medians are given.
Here's a minimalist statement of a problem I just came up with :
There is a triangle $\triangle ABC$ with $\angle C=73^\circ$. And $AF=2BE$ when $E, F$ are midpoints of $AC,BC$ respectively.
My ...
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What is the measure of angle $A$, given $|AB|$ and $|AC|$ , such that the incircle of $\triangle ABC$ is maximal?
Here's a problem I just came up with:
I considered a triangle $\triangle ABC$ such $|AB|=11$ and $|AC|=5$.
I asked myself the following question:
What must angle A be when the incircle of ABC has a ...
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Prove that $\angle POB = 2\angle PBO$
The point $A$ is on circle with centre $O$. $OA$ is extended to $C$ s.t. $OA=AC$, and $B$ is the midpoint of $AC$. The point $Q$ is on the circle such that $\angle AOQ$ is obtuse. The line QO meets ...
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Prove that $F$ is a point such that $PAEF$ is a rectangle then $PF$ bisects $\angle BFC$
The point $P$ lies on the internal bisector of $\angle BAC$. The point $D$ is the midpoint of $BC$ and $PD$ meets the external bisector of $\angle BAC$ at $E$.
Prove that if $F$ is the point such that ...
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Find the minimum value of $\frac{BX}{FC}$ and find the triangles $\triangle ABC$ when it is possible
In triangle $\triangle ABC$, the points $L, M$ are the midpoints of $BC$ and $CA$, respectively, and $CF$ is the altitude from $C.$ The circle through $A$ and $M$ which $AL$ is tangent to at $A$ meets ...
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Is the old radian symbol (a superscript "c") now defunct?
At school in 1975, we were introduced to measuring angles in radians.
The symbol used then was a sort of small c written as a superscript, e.g. as in the expression below
$$ \theta \; = \; \dfrac{\pi}{...
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An angle in a triangle satisfying $a^3 + b^3 = c^3$
$\triangle ABC$ is a triangle. Let $|BC|= a$ ; $|AC|= b$ ; $|AB|= c$.
We know that if: $a^2 + b^2 = c^2$, then angle $C$ is equal to $90$°. So, I asked myself the following question:
What happens when,...
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Find the angle ADB
Let ADE be a triangle with $\angle DEA = 90^\circ$ and $\angle DAE = 75^\circ$. Let B be a point inside the triangle such that $\angle DEB = 15^\circ$ and $\angle DAB = 45^\circ$. Find the measure of ...
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How to find dihedral angle in a regular right pyramid with polygonal base?
Consider a regular right pyramid with polygonal base having n number of sides. IF $\lambda$ is the angle between the polygonal base and lateral face APN in pyramid, what will be the dihedral angle $\...
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Confusion about interior vs. exterior angles in a regular octagon diagram
This is the Question with diagram from the book:
I am solving the problem using the following theorems:
Theorem: Polygon Angle-Sum Theorem:
The sum of the measures of the interior angles of an n-gon ...
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Why is the incenter needed to prove that $CN$ bisects $\angle MNE$?
Problem: Let $ABC$ be a triangle inscribed in a circle $\omega$ with $AB > AC$. Let $M$ be the midpoint of $BC$ (so $BM = MC$). Let $AM$ intersect $\omega$ at point $D$. Let $E$ be a point on $\...
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Prove that $HD=AE$ if $\triangle ABC$ is an acute scalene triangle
Let $\triangle ABC$ be an acute scalene triangle with incenter $I$ and orthocenter $H$. Let $M$ be the midpoint of $\overline{AB}$. On the line $\overline{AH}$, consider points $D$ and $E$ such that $\...
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Solving fifth-order polynomial from dividing angle into five
I know that, for angle trisection, if
$a = \displaystyle\tan\frac{\arctan\frac{1}{2}}{3}$
then
$a = \displaystyle\frac{1-\left(\frac{1-i\sqrt{3}}{2}\right)\sqrt[3]{5-10i}-\left(\frac{1+i\sqrt{3}}{2}\...
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Finding $\frac{AH}{HG}$ given $\angle BAG =120°$, $F$ midpoint of $AB$, and $\angle AFH = \angle HBG$
In $\triangle ABG$,$AB=AG$, $\angle BAG=120^{\circ}$. $F$ is the midpoint of $\overline{AB}$. $H$ lies on $\overline{AG}$ such that $\angle AFH=\angle HBG$. Compute $\frac{AH}{HG}$.
Initially, I ...
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