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Questions tagged [euclidean-geometry]

For questions on geometry assuming Euclid's parallel postulate.

The geometry of Euclid is based on five axioms (Euclid called them postulates). Any geometry based on the first four of these is called an absolute geometry. The fifth one states:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

It was observed by Proclus that, in the presence of the the other four postulates, Euclid's fifth postulate can be replaced by Playfair's axiom:

Given a line and a point not on it, then one and only one line parallel to the given line can be drawn through the point.

The independence of the parallel postulate and its equivalent formulations from the first four axioms was shown by Beltrami in 1868.

Another alternative definition is that two lines are parallel if every perpendicular extended from one meets the other as a perpendicular.

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Rhombus and circles angle problem

Let $ABCD$ be a rhombus. The circle $(C_1)$ of center $B$ passing through $C$ and the circle $(C_2)$ of center $C$ passing through $B$. $E$ is one of the two points of $(C_1) \cap (C_2)$. The line $(ED)$ meets $(C_1)$ again in $F$. It is asked to…
ahmed
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Geometry - Rectangle ABCD with inside point E. Find the least possible value for sum of interger distances from E to 4 vertices.

The point E lies within the rectangle ABCD. If the distances from the vertices to E are all distinct integers, what is the least possible value of AE + BE + CE + DE?
YEET
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Prove that $MD$ passes through the midpoint of $IE$.

$E$ is the midpoint of line segments $BC$. $F$ is the reflection of $E$ in $(A, B, C)$. $EF \cap (A, B, C) = M$. $I$ is the incentre of $\triangle{ABC}$. $AF \cap (A, B, C) = G$, $GI \cap (A, B, C) = D$ ($D \not\equiv G \not\equiv A$). Prove that…
Lê Thành Đạt
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Construct a pentagon from the midpoints of its sides

Let $p_{1},p_{2},p_{3},p_{4},p_{5}$ be five points in the euclidean plane such that no set of three of those points lie on the same line. It is easy to prove that there exists a unique pentagon such that $p_{1},p_{2},p_{3},p_{4},p_{5}$ are the…
Vincent Pfenninger
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A criterion to obtain the feet of two altitudes of an acute triangle

$\triangle{ABC}$ is an acute triangle, and $R$ is the foot of the altitude from $C$. If $H$ and $K$ are the reflections of $R$ across $\overline{\mathit{BC}}$ and $\overline{\mathit{AC}}$, respectively, and if $P$ and $Q$ are the intersections of…
Adelyn
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How can we construct rectangle with non-zero area when all its lines have zero area?

[Content was updated after discussion with David C. Ullrich] I have one method of construction of rectangle, but it gives incorrect results. The reasoning for this faulty method is following: Consider this rectangle: As you can see I divided it…
KarmaPeasant
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A property of centroid

Let $\triangle ABC$ be an arbitrary triangle and let $G$ be its centroid. Three medians are denoted by $AD,BE,CF$. I am attempting to show that the circumcentres of $\triangle AGF,\triangle GFB,\triangle BGD,\triangle DGC,\triangle CGE,\triangle…
Phil. Z
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Prove IZ is perpendicular to AN

Given $\Delta ABC$. A circle pass through $B,C$ intersects $AC,AB$ at $E.F$. Let $N$ be the midpoint of $EF$. $(ANE) \cap AB=X, (ANF) \cap AB=Y$. A line through $A$ and parallel to $BC$ intersects $EF$ at $Z$. Let $I$ be the center of $(AXY)$. Prove…
RopuToran
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Dividing a paper into 3 parts using folding

So I saw this video : https://www.youtube.com/watch?v=V7UJxeMlowQ, and it got me wondering: Why does that method divide the paper into exactly 3 equal rectangles, so I turned it into a math problem: Draw a line from one vertex of a square to the…
pavle
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Prove the point is the midpoint of a segment

Let a circle (O) and a point A outside of the circle. AB, AC are tangents of (O) (B,C $\in$ (O)). BD is an diameter of (O). CK perpendicular to BD (K $\in$ BD). Let I is intersection of CK and AD. Prove that I is midpoint of KC
Curiosity
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Find measure of angles

Let $ABCD$ be a convex quadrilateral and $M$ be the midpoint of the segment $BC$ such that $$∠AMD = 90^\circ,\ ∠ADM = 15^\circ,$$ and $$AD = AB + CD.$$ Find $∠BAD$. I try to find an additional construction but I don't know. Can you help me with…
Problemsolving
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Bound for perimeter of convex polygon?

Is it true that the perimeter of any convex polygon in the unit disk on the Euclidean plane is less than the circumference $2\pi$ of the circle? Thanks. [The OP has solved it]
mathfan
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How many distances are required to find a point in an n-dimensional Euclidean space?

I am not a mathematician and I apologize if this question is naive. I am interested in knowing how many distances are needed to determine the coordinates of a point in a Euclidean space of $n$ dimensions. I know no more than 3 distances are needed…
Michele
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Menelaus's Theorem clarification

If three points, one on each side of a triangle (extended if necessary) are collinear, then the product of the ratios of division of the sides by the points is -1 if internal ratios are considered positive and external ratios are considered…
Lemon
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How would you solve this geometry problem? (MMC $2015$)

Question: Find the area of the shaded region given $EB=2,CD=3,BC=10$ and $\angle EBC=\angle BCD=90^{\circ}$. I first dropped an altitude from $A$ to $BC$ forming two cases of similar triangles. Let the point where the altitude meets $BC$ be $X$.…
Frank
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