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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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Why does having alternate interior angles congruent, etc., prove that two lines are parallel?

I know that if two lines are parallel and there is a transversal crossing both, the alternate interior angles are congruent, alternate exterior angles congruent, etc. etc. According to the geometry textbook that the student I'm tutoring brought, the…
DonielF
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Equal sums of surfaces

The inner points $X$ and $Y$ in the convex quadrilateral $ABCD$ are such that $\angle ABX=\angle CBY$, $\angle BCY=\angle DCX$, $\angle CDY=\angle ADX$ and $\angle DAY=\angle BAX$. Prove that $$S^{}_{AXB}+S^{}_{CXD}=S^{}_{AYD}+S^{}_{BYC}$$
J. Holtan
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Cross Products vs using geometry

Say I wanted to calculate the area of a figure comprised of the coordinates (0,1,1) (1,0,1) and (0,0,0). When I used cross products I get the area as the square root of 3/2, however, when I use standard geometry I get the answer as 1/6. Can someone…
user501887
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Correlation between parallel lines and volumes of tetrahedrons.

Given 4 parallel lines $d_1$, $d_2$, $d_3$, $d_4$ , no more than 2 of which can be on a same plane. Plane (P) intersects the 4 lines at 4 points A, B, C, D. Plane (Q) ( not identical to plane (P)) intersects the 4 lines at $A_1$, $B_1$, $C_1$,…
Nguyen Xuan Minh
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Proving colinearity of 3 points(basic Euclidean geometry)

In the above image, AP is bisector of the angle BAC and DP is a perpendicular bisector of the segment BC. How can it be proved that the points E, D, F are colinear?
Kiyoon Eum
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Angle trisection using a Cardioid

How does one proceed to trisect an angle using the cardioid? It is known that Etienne Pascal, father of Blaise Pascal, has devised a way to do it. I haven´t found the method in my search on the literature. Best Regards.
Petruchio de São Zeno
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Stuck with Euclidean space geometry exercise

$ABCD$ is a tetrahedron. $E$ is a point on segment $AD$. $Δ$ is a line of plane $(BCD)$ parallel to $(BC)$. Line $Δ$ intersects $(BD)$ at $I$ and $(CD)$ at $J$. Plane $(ABC)$ intersects line $(EI)$ at $M$ and line $(EJ)$ at $N$. Show that lines…
fgrieu
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Proving two circle chords congruent

My kid has this homework question: Two circles whose centers are $M$ and $N$ meet at points $A$ and $B$. The point $N$ is on the circle whose center is $M$. The tangent at $A$ to the circle whose center is $M$ meets the circle whose center is $N$…
msh210
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Prove that if an affine transformation $T : \mathbb{R}^2 \to \mathbb{R}^2$ is onto, then it is one-to-one.

Prove that if an affine transformation $T : \mathbb{R}^2 \to \mathbb{R}^2$ is onto, then it is one-to-one. Here we define an affine transformation as $T:\mathbb{R}^2 \to \mathbb{R}^2$ s.t. $T(r_1A_1 + ... + r_nA_n) = r_1T(A_1) + ... + r_nT(A_n)$ for…
frostyfeet
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Show that KPMQ is Cylic

Let AB be a diameter of a circle C, M a point on the circle, K a point on the Diameter. If P and Q are the circumcenters of the triangles AMK and BMK, show that the KPMQ is cyclic. I have tried for quite a while now, but all I can seem to get is…
Pepe Pérez
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Congruence of angles with vertex at Fermat-Torricelli point through any two vertices

At the following website, $\angle{ARF} \cong \angle{ABF}$ is asserted. https://en.wikipedia.org/wiki/Fermat_point#Construction (See section labeled Location of X(13).) The Inscribed-Angle Theorem is tacitly used. $A$, $B$, and $R$ are, by…
Adelyn
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Barycentric coordinates of the centroid of parallelogram

Let ABCD be a non-collinear parallelogram, E be the midpoint of AB, and F be the midpoint of BC. Prove that D, E, F form an affine basis, and find the barycentric coordinates of the centroid of A, B, C, D with respect to D, E, F. I am able to show…
frostyfeet
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Problem with angles in an isosceles triangle

Given an isosceles $ABC$ triangle ($AB=AC$), and points $D,E$ on sides $AC, AB$, respectively, such that $AD
user173628192
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Prove angle AGN = angle PGM'

Given $\Delta ABC$ with its altitudes $AD,BE,CF$ and orthocenter $H$. Let $M,N,P$ the midpoint of $BC,AH,EF$. Let $G$ be the foot of perpendicular from $A$ to $EF$. Let $M'$ be the image of $M$ by reflection wrt $D$-midline of $\Delta DEF$. Prove…
RopuToran
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Prove angle IPM=90

Given triangle $ABC$ inscribed $(O)$. Let $I$ be the incenter and $D$ be the contact point of $(I)$ with $BC$. $AD$ intersect $(O)$ at the second point $E$. Let $M$ be the midpoint of $BC$ and $N$ the midpoint of arc $BAC$. Let $EN$ intersect…
RopuToran
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