# Questions tagged [euclidean-geometry]

Geometry assuming the parallel postulate: in a plane, given a line and a point not on that line, there is exactly one line parallel to the given line through the given point.

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### A problem concerning a parallelogram and a circle

Sorry for the ambiguous title. If you can phrase it better, feel free to edit. "A parallelogram $ABCD$ has sides $AB = 16$ and $AD = 20$. A circle, which passes through the point $C$, touches the ...
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### Geometry question: Find the area of blue-shared area inside this isosceles

See below, looks a bit interesting, but I cannot find a solution. I think a starting point might be the similarity of the lower white triangle and the larger triangle composed of lower blue, pink, and ...
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### Show one diagonal of $B D E C$ divides the other diagonal in the ratio?

Consider a triangle $A B C$. The sides $A B$ and $A C$ are extended to points $D$ and $E,$ respectively, such that $A D=3 A B$ and $A B=3 A C$. Then one diagonal of $B D E C$ divides the other ...
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### $3$ points are collinear

$M$ is a point on $AB$. $BMC$ and $AMD$ are constructed such that they are both equilateral & on the same side of $AB$. The circumcircle of both triangle intersect at $M$ and $N$. Prove that $AD$ ...
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### Suppose $AD$ is a height of $\triangle ABC$ and $H$ is the orthocenter. Is it true that $BD \cdot DC = AD \cdot DH$?

Suppose $AD$ is a height of $\triangle ABC$ and $H$ is the orthocenter. Is it true that $BD \cdot DC = AD \cdot DH$? I sense that it is false and I have tried to find a counterexample, but I cannot ...
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### Property of cyclic quadrilaterals [closed]

Suppose $ABCD$ is a cyclic quadrilateral and $P$ is the intersection of the lines determined by $AB$ and $CD$. Show that $PA·PB= PD·PC$ Could you help me please, I have no idea how to relate the ...
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### Prove BMXN is cyclic.

Suppose $C_1$ and $C_2$ are circles such that {$𝐴,𝐵$}=$𝐶_{1}\cap 𝐶_2$. We draw a secant $MN$ such that $𝑀\in 𝐶_1$ and $𝑁\in 𝐶_2$, and $A\in MN$. Show that if $X$ is the point of intersection ...
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### Crux problem #33 with vector approach

On the sides $CA$ and $CB$ of an isosceles right-angled triangle $ABC$, points $D$ and $E$ are chosen such that $|CD|=|CE|$. The perpendiculars from $D$ and $C$ on $AE$ intersect the hypotenuse $AB$ ...
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### One altitude, one bisector and one median are equal implies equilateral triangle

Let $ABC$ be a triangle such that the lenghts of the $A$-angle-bisector, the $B$-median and the $C$-altitude are equal. Prove that $ABC$ is equilateral. I was only able to show that $AB$ is the ...
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### Area of Projection to a Plane

I need to find all planes of the form $ax+by+cz=0$ such that the projection of: $$S=\left\{x^2+y^2+z^2=4\middle|x^2+y^2\leq 2x\right\}$$ Onto that plane is one-to-one, and then use it somehow to ...
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### Expected maximum sum of pairwise distance for *n* points on a cylinder

I'm trying to calculate a measure of 'concentration' or 'clusteredness' for n points on a cylinder. Say I start with a plane of width=12 (x) and height=10 (y) on which 10 points are situated. These ...
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### Proving inequality of line segments in a circle

There are 2 intersecting circles with the centers $O_1$ and $O_2$ and the radiuses $r_1$ and $r_2$ respectively ($r_1 \gt r_2$). They have a common segment line $AB$. Also $AC$ is the tangent line of ...
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### Maximum number of acute triangles in a polygon convex

If $𝑃$ is a convex polygon and is divided into $𝑛 − 2$ triangles with diagonals. What is the maximum number of acute triangles you can have? I did not understand the part of triangles with diagonals,...
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### RMO 1990 question [closed]

Prove that the inradius of a right angled triangle having integer sides is also integral I tried it and got something like $r=\frac {(a.b)}{(a+b+c)}$ How to proceed after this.
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