Mathematics Stack Exchange
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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.

9328 questions
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How to find an opening angle in a concave quadrilateral polygon?

The problem is as follows: In the figure find $\angle{ABC}$. It is known $AB=BC=AD$. The choices given in my book…
Chris Steinbeck Bell
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How to find the altitude of a right triangle given angles and a sum of their sides?

The problem is as follows: In the figure it is known $VN=NA$ and $AH+HM=MV$ and $VA=10\,cm$. Using this information find $AH$. The choices given in my book are: $\begin{array}{cc} 1.&\textrm{6 cm}\\ 2.&\textrm{7 cm}\\ 3.&\textrm{8…
Chris Steinbeck Bell
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How to find the measure of an angle on the interior of an isosceles triangle?

The problem is as follows: On the interior of an isosceles triangle $\triangle{ABC}$ where $\angle{B}=110^{\circ}$ it is situated a point $M$ such as $AB=MC$ and $\angle{BAM}=5^{\circ}$. Using this information find $\angle{MCA}$. The choices in my…
Chris Steinbeck Bell
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How to find the angle and double angle in a triangle given their lengths?

The problem is as follows: In the figure, $AC=2AB$. Using this information find $\alpha$. The choices given in my book…
Chris Steinbeck Bell
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Reciprocal of midsegment (midpoint) theorem proof

I need a hint on how to prove that "In a triangle ABC if D is the midpoint of the side AB and the line through D intersects AC at E such that BC=2DE then E is the midpoint of AC and BC is parallel to DE" Thank you
rainthree
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Prove that the sum of areas of triangles $AOH$ and $BOH$ equals the area of triangle $COH$.

Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Prove that the area of one of the triangles $AOH, BOH,$ and $COH$ is equal to the sum of the areas of the other two. In this figure, we want to prove $[AOH]+ [BOH]=…
PNT
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What is the cardinality of the set of shapes?

Consider the Euclidean plane $\mathbb{R}^2$. A figure is a subset of the Euclidean plane. Two figures $S$ and $T$ are said to have the same shape iff there is a composition of an isometry and a dilation on $\mathbb{R}^2$ such that the image of $S$…
user107952
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Find $\angle{CAD}$ in equilateral triangle $\triangle{ABC}$ for $D$ inside with $\angle{ABD}=18^{\circ}, \angle{BCD}=12^{\circ}$

This problem is feasible to solve using trigonometric functions. I am looking for pure geometric solution as usual. Thanks...
r ne
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Simson's line proof

Q. Let $ABC$ be a triangle and P be any point on $(ABC)$. Let X, Y , Z be the feet of the perpendiculars from P onto lines BC, CA, and AB respectively. Prove that points $X, Y , Z$ are collinear. Actually I started going through the book EGMO and…
Sauhardya Hazra
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Why is $a\cong b$ in this problem?

I am trying to understand the following problem: State the basic angle theorem needed to prove $a\cong b$ in the following figure: The answer is: Since $\overline{AB} \perp \overline{BC}$, $B$ is a right angle and hence, $b$ is the complement of…
Red Banana
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How to prove that the midpoints in a square generate a one twentieth triangle area?

I found this identity or sort of in my book, but there isn't a proof of this. The figure is attached below. As it can be seen it is a square where it has been traced four lines, going from all their corners to the midpoints of their opposing…
Chris Steinbeck Bell
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Is there a geometric way to find the normal to a smooth surface in R3

Given S -- a smooth surface in 3 dimensional space, p in S. Needham doesn't really define things like curves and surfaces. He takes the Euclidean approach that you know what they are. So smooth will have to do as a description of the surface. In…
lewis robinson
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Extend Ceva's theorem to angles?

Ceva's theorem, as described in wiki Given a triangle ABC, let the lines AO, BO and CO be drawn from the vertices to a common point O (not on one of the sides of ABC), to meet opposite sides at D, E and F respectively. (The segments AD, BE, and CF…
athos
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How to order 8 vertices that are known to belong to a cuboid?

Given 8 points in $\mathbb{R}^3$ and the prior knowledge that these points form the vertices of a cuboid, what is a computationally efficient way to order them? When I say "cuboid" I refer to the set of all achievable shapes when starting from the…
Alexander Soare
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Mass Points: When does the centroid of the triangle formed by the cevians coincide with the original triangle?

I'm studying mass points and I stumbled upon this problem: "Consider $\Delta ABC$ with points $P, Q, R$ on $AB, BC, AC$ with $AP:PB=BQ:QC = CR:RA$. Then the centroid of the triangle formed by $AQ, BR, CP$ coincides with the centroid of the original…
Misha Shklyar
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