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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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Dot Product vs. Product of Norms

I was working on an algebraic proof for the Triangle Inequality when I came across an instance in which $$(a \cdot b) \le ||a|| \cdot ||b||$$ Is this true in general, or was this just a special case? Thanks in advance!
bjorn
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Calculating an equilateral triangle side length

I have an equation that breaks down a relationship between the side lengths of $\frac{1}{2}$ of an equilateral triangle. I there is a sub step that I do not understand, I will list all steps for completeness. The step that I do not understand is…
GettingItDone
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Find Point such that vectors are perpendicular

Given three points P,Q and R, how May I find another point O such that OP,OQ and OR are all perpendicular between themselves? I have no idea how to tackle this
user10000024
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minimization problem in the set of isosceles right triangles

Let $ABC$ be a right angle triangle with $BC = AC =1$. let $P$ be any point on $AB$. Draw perpendiculars $PQ$ and $PR$ on $AC$ and $BC$ respectively from $P$. Define $M$ to be the maximum of the areas of $BPR$, $APQ$ and $PQCR$. Find the minimum…
MathIsMyPassion
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$ABC$ is a triangle and $A',B',C'$ are the midpoints of the sides $BC,CA,AB$ respectively.If $AD$ is the altitude through $A$, prove $B'DA'=BCA.$

Now, the question is fairly specific, $ABC$ is a triangle, and $AD$ is an altitude. Midpoints of the sides being $A'$ for $BC$,$B'$ for $CA$ and $C$' for $AB$. Edit: By similarity I proved the
A Guru
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Draw geometric shapes dividing n-dimensional space in sectors closer to selected points

Let's say I am on a 2-dimensional space, and I have two points $(x_0, y_0)$ and $(x_1, y_1)$. To divide the plane into two parts, the set of points closer to $(x_0, y_0)$ and the set of points closer to $(x_1, y_1)$ (with the usual metric), I simply…
user
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Formula for sum of interior angles in polygon

I am reading Aigner & Ziegler's Proofs from the BOOK Chapter 35 and came across the following statement: The astute reader may have noticed a subtle point in our reasoning. Does a triangulation [of a polygon] always exist? Probably everybody’s…
Earthliŋ
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How to find line segment length $BC$ in angle $BAC$?

So I'm getting frustrated with this problem I'm having with this geometry problem. I have an angle $BAC$, and line segments $AB$ and $AC$ are equal to $1$ unit long. Given the angle $BAC$, what is the length of the line segment $BC$?
BrodieBooYT
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Find angle in triangle with given perimeter

Given triangle $ABC$ with perimeter $P=25$. If $O$ is the center of inscribed circle and $CO=4$, prove that angle $C=60^0$.
kmitov
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Midpoints of parallel chords in an ellipse

One of many properties of an ellipse is this: the locus of midpoints of parallel chords is a straight line. A proof I present below is more or less brute-force. I am looking for a purely geometric proof, using perhaps the very definition of the…
uniquesolution
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In the figure, OB is the perpendicular bisector of line segment DE

In figure, $OB$ is perpendicular bisector of line segment $DE$, $FA$ perpendicular to $OB$ and $FE$ intersects $OB$ at the point $C$, then find the sum $\displaystyle \frac{OC}{OA}+\frac{OC}{OB}$ I have proved the similarity of the triangles FAC~EBC…
Mohit Kumar
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Eight point circle: why are all conditions met, but the points are not as expected?

I was trying to draw the eight point circle for a convex quadrilateral. The only condition that I know of is that it must be a convex quadrilateral with perpendicular diagonals. This is a picture of a successful construction using GeoGebra. I…
Cheng
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How the information about acute angles be used to find the length in a segment of a triangle?

The problem is as follows: In an acute angled triangle $ABC$ $(AB
Chris Steinbeck Bell
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How to get the angle in a trapezoid formed by two triangles?

The problem is as follows: Find the value of angle $\angle BCA$ or $x$. The alternatives given in my book are as follows: $\begin{array}{ll} 1.&30^\circ\\ 2.&20^\circ\\ 3.&10^\circ\\ 4.&40^\circ\\ \end{array}$ The thing here is without any further…
Chris Steinbeck Bell
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How to find the angle formed below a triangle joined by another?

The alternatives given in my book are as follows: $\begin{array}{ll} 1.&30^{\circ}\\ 2.&40^{\circ}\\ 3.&20^{\circ}\\ 4.&45^{\circ}\\ \end{array}$ I'm stuck with this problem. What exactly should it be done here?. The thing is that I was only able…
Chris Steinbeck Bell
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