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.

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How to find the length of a segment in a quadrilateral made by two triangles?

The problem is as follows: The alternatives given in my book are as follows: $\begin{array}{ll} 1.&\textrm{3 cm}\\ 2.&\textrm{4 cm}\\ 3.&\textrm{5 cm}\\ 4.&\textrm{6 cm}\\ \end{array}$ I'm stuck at trying to find the proper relationships in this…
Chris Steinbeck Bell
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Geometric problem concerning relation of equal segments and angles

Triangle ABC Suppose that $P$ and $Q$ are points on the sides $AB$ and $AC$ respectively of $△ABC$. The perpendiculars to the sides $AB$ and $AC$ at $P$ and $Q$ respectively meet at $D$, an interior point of $△ABC$. If $M$ is the midpoint of $BC$,…
Limestone
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The area of a square with length $\pi$ is $\pi^2$ by definition?

Area is the quantity that expresses the extent of a two-dimensional figure or shape or planar lamina, in the plane. When its sides have lengths of irrational numbers, are their areas defined, for example, the area of a square with length $\pi$ is…
user824904
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Does it exist a way to find one side of a triangle in polygon other than using similarity?

The problem is as follows: Using the figure from below find $EB$. Assume $\triangle ABC$ is isosceles where $AB=AC$ and $\overline{BN} \parallel \overline{AC}$. Find $EB$. The alternatives given in my book are as…
Chris Steinbeck Bell
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If $||x||_2 = 3$ and $||y||_2 = 7$ find the minimum of $\langle x, y\rangle$

If $||x||_2 = 3$ and $||y||_2 = 7$, find the minimum of $\langle x, y\rangle$ My thought process is that it would be 3 since if $||x||_2 = 3$, we can have $|x_1| = 1$, $|x_2| = 1$, $|x_3| = 1$ and every other term can equal $0$. Similarly, if…
Nuemann12
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Euclidean Norm Calculation

If ||x||$_2$ = 3, ||4x-5y||$_2$ = 14 and ||5x+2y||$_2$ = 8 find ||y||$_2$ I had attempted to write at the vectors in explicit form and in doing so arrived at a point where I would add (||4x-5y||$_2$)$^2$ = (14)$^2$ and (2)(||5x+2y||$_2$)$^2$=…
Nuemann12
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How to find the cost to cover with a fence a side of a triangle?

The problem is as follows: The figure from below shows a certain terrain has the shape of a triangle $\triangle ABC$ which is obtuse on $B$. Assuming the line connecting $AM$ is a bisector whose angle is $\beta$ and $\angle ACB=20^{\circ}$ and…
Chris Steinbeck Bell
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How to find the length of one segment in a triangle when it is crossed by another triangle?

The problem is as follows: The figure from below shows a parallelogram $ABCD$ and $\triangle\,APD$ and $QR=3\,cm\,RD=4\,cm$. Using this information find the value of PQ. The alternatives given in my book are as…
Chris Steinbeck Bell
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How to find the angle between two triangles when there is involved the sum of two sides not adjacent?

The problem is as follows: The figure from below shows a triangle $ABC$. Find the angle $x$ on $\angle BCA$. The alternatives given in my book are as…
Chris Steinbeck Bell
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How to find the reduced angle of a triangle when the interior ones are given as multiples?

The problem is as follows: The figure from below shows a quadrilateral. Find the angle labeled $x$. The alternatives given in my book are as…
Chris Steinbeck Bell
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Deducing the Pythagorean Theorem from a particular dissection

How can I deduce the pythagorean theorem from the follow image? I have been draw some parallels and I got the figure but I don't know how to deduct, some hint?
Kevin Duran
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Algebraic derivation for expression $2r \sin((\Theta_2 - \Theta_1)/2)$ for chord of circle of radius r subtending angle $\Theta_2 - \Theta_1$.

This diagram: indicates how the expression $2r\sin((\Theta_2 - \Theta_1)/2)$ for chord of circle of radius r subtending angle $\Theta_2 - \Theta_1$ can be validated geometrically (see that the segment n is half of $\delta$, where $\delta$ is $d(P1,…
user77970
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Prove that a line drawn from a point on a circle to another point inside will intersect the circle exactly one more time.

I have so far... Given a line l drawn from a point A on the circle to another point B on the inside of the circle. We want to show that line l will intersect the circle exactly one more time. I know I need to use theorem 2.25: Given a circle and a…
Lacey Schleicher
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Prove equilateral triangle in another equilateral triangle

As shown above, given $AB=BC=AC$ and $BE=CF=AD$, to prove $DF=FE=ED$. (there's a typo on the image) Furthermore, is such property true for any regular $n$-sided polygon？
athos
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Why is 1.4 necessary in Euclid's Proposition 24?

In proposition 24 (1.24) of Elements, Euclid uses proposition 4 (1.4) to demonstrate triangle equality before going on to finish the proof. However, before he uses 1.4, he uses proposition 23 (1.23) to replicate an angle of one of the given…
LRoseR
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