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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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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Prove that $D$ is the incenter of $XH'Y$.
Problem:
Let $ABC$ be inscribed in a circle. Let $D \in BC$ and $AD\perp BC$. Let $M \in AB$, $N \in AC$ such that $DM \perp AB$ and $DN \perp AC$. $MN$ intersects the circumcircle at $X$ and $Y$. $...
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Prove triangle $\triangle AED$ is isosceles if the tangents of point $P$ outside circle $ABC$ are drawn
Let $P$ be a point lying outside of a circle $O$. A line through $P$ is tangent to $O$ at A. A second line through $P$ intersects $O$ at two distinct points $B$ and $C$ such that point $B$ lies ...
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AMC 10 Mock Test Geometry Problem: why is the answer 110 and not 140? [closed]
Let $\triangle PQR$ be an isosceles triangle with $QR = PR$ and $\angle QRP = 40°$. Construct a circle with diameter $QR$, and let $S$ and $T$ be the other intersection points of the circle with $PR$ ...
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Show that $AR\perp AO$
Let a acute $\triangle ABC$ (neither right nor isosceles, $AB<AC$) be inscribed in a circle $(O)$. Let $E,F$ satisfies $\widehat{ABE}=\widehat{CAE}=\widehat{BAF}=\widehat{ACF}=90^\circ$. I want to ...
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What are the measures of the acute angles of this right triangle?
In this figure of the plane of the right triangle OAB in which we have noted the necessary indications, we ask for the measures of the acute angles of this triangle .
With GeoGebra, my answer is 31° ...
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Finding the interior and exterior angles of a regular nonagon
Quadrilateral angle sum:
I know that this would have to equal 360 degrees.
$\angle A=90, \angle C=140$ so the angle inside the quadrilateral is $360-140=220$ but I'm not sure how to get $\angle ABC$
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Prove isosceles right triangle $\triangle PNQ$ with angle chasing
Let $AB$ be the diameter of a circle $O$. If tangent $CN$ is such that $C$ lies on line $AB$, and the angle bisector of $\angle NCA$ intersects $NA$ at $P$, and $NB$ at $Q$,
Prove that $NQ=NP$.
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Find the measure of $∠OTS$ in a circle
I need help with the following Geometry problem.
In a circle with diameter $AB$ and center $O$, chords $PQ$ and $QC$ are drawn that intersect $AB$ at $M$ and $N$, respectively. If the extensions of $...
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What is the exact measure of this angle?
In the attached figure where points $A$, $B$, and $C$ are collinear, the lengths of $AE$, $EB$, $BD$, and $DC$ are indicated. We are asked for the exact value of angle $\hat A$.
With GeoGebra, I found ...
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When four lines form obtuse triangles in every triple, must their obtuse sectors have non-empty intersection?
Suppose a pair of lines bounds two angles at their intersections; one acute and one obtuse.
We call the obtuse sector the region of the plane inside the larger of the two angles formed by the two ...
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Show that $\widehat{MKI}=90^\circ $
Let $\triangle ABC$ (not isosceles or right at $A$) be inscribed in a circle $(O)$, $I$ is midpoint of $BC$ and $H\in BC, E\in AC, F\in AB$ such that $AH\perp BC,\, BE\perp AC,\, CF\perp AB$. Let $K\...
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Angle Bisectors and a Parallelogram Problem
Here is a problem:
$ABCD$ is a parallelogram with $\lambda={AB/BC}$. Plot an arbitrary point $P$ on line segment $CD$. Produce $CD$ such that $AD$ bisects angle$PAQ$. Plot $M$ on $BC$ such that $AM$ ...
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What is the measure of angle ACB in this simple configuration?
Here's a short question from my old book :
$G$ is the centroid of $\triangle ABC$ such that angle $\angle ACG$ measures $25^\circ$. What is the the measure of angle $\angle ACB$ in each of the ...
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Angle computation using trigonometry
I have a camera-light setup, where I know following parameters
Vertical angle of camera: 25°
Height of camera: 270 mm
Height of illuminator: 190 mm
Distance between camera and illuminator: 200 mm
...
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Under the conditions given below, calculate the angle $ABP$ of the triangle $ABC$.
In $\triangle ABC$ a point $P$ is located external to side $AC$ such that $BC = CP = PA.$ $\space$ If $m∠BAC = 75^\circ \text{ and } m∠BCP = 90^\circ$ then find $m∠ABP. \space$ (The answer is supposed ...
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A confusing proof of orthocentric triangle
I found orthocentric triangle proof really confusing at angles part. There isn't really much online info about proving such a triangle, so there is no valid explanation of the process
This particular ...
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How to find the central angles in a spherical pyramid given its spherical triangle base?
Given $\alpha$, $\beta$, and $\gamma$, or even just knowing the excess of the spherical triangle, is there a way to calculate the central angles $\angle AOB$, $\angle AOC$, and $\angle BOC$ in order ...
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Proof of the identity: $\frac{\tan(70^\circ) - \tan(60^\circ)}{1 - \frac{\tan(70^\circ) \cdot \tan(60^\circ)}{\tan^2(80^\circ)}} = \tan(50^\circ)$
While browsing the comment section of a math video on YouTube, I came across this curious trigonometric expression:
$$
\frac{\tan(70^\circ) - \tan(60^\circ)}{1 - \frac{\tan(70^\circ) \cdot \tan(60^\...
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Can side lengths and a single angle uniquely determine a polygon?
I came across a problem on Reddit where all five side lengths of a pentagon were given, along with one interior angle (90°). That got me wondering:
Is there a general relationship between the side ...
user1663226
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Trying to Prove Converse of Euclid Book 3 Proposition 21
Euclid's Elements, Book 3, Proposition 21 states that in any circle, any two inscribed angles on the same arc will be equal to each other. In other words, given a chord $\overline{AB}$ and two other ...
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In $\triangle ABC$, prove that $IE^2 = BE \cdot CE$.
In $\triangle ABC$, let $I$ be the incenter, and let $D$ be the midpoints of the arc $BAC$ on the circumcircle. The line $DI$ intersects the circumcircle again at $E$. Prove that $IE^2 = BE \cdot CE$.
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Name of curve or spiral such that there is a point with constant angle to tangent going through that point
A circle has the property that there is a point that all normal lines (i.e. $90^{\circ}$ anticlockwise to the tangent) go through that point. [In fact, I think a circle is the only curve with this ...
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Prove that $BD=CF$ where we use the incenter of $\triangle ABC$
Let $I$ be the incenter of $\triangle ABC$, where it touches $BC$ at $D$. Let $DE$ be the diameter of the incircle, and let line $AE$ intersect $BC$ at $F$.
Prove that $BD=CF$.
My attempt was using ...
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