All Questions
Tagged with triangles euclidean-geometry
46
questions
1
vote
0
answers
42
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Triangle centers formed a rectangle associated with a convex cyclic quadrilateral
Similarly Japanese theorem for cyclic quadrilaterals, Napoleon theorem, Thébault's theorem, I found a result as follows and I am looking for a proof that:
Let $ABCD$ be a convex cyclic quadrilateral.
...
- 1,478
4
votes
3
answers
974
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Is there a pyramid with all four faces being right triangles? [closed]
If such a pyramid exists, could someone provide the coordinates of its vertices?
2
votes
1
answer
148
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Finding angle with geometric approach [closed]
I would like to solve the problem in this picture:
with just an elementary geometric approach. I already solved with trigonometry, e.g. using the Bretschneider formula, finding that the angle $ x = ...
- 29
6
votes
0
answers
117
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How many equilaterals have vertices intersections of angle trisectors of a triangle?
The celebrated Morley’s theorem ensures that the interior trisectors, proximal to sides respectively, meet at vertices of an equilateral.
In the paper Trisectors like Bisectors with Equilaterals ...
16
votes
1
answer
435
views
Is there a conceptual reason why so many triplets of lines in a triangle are concurrent?
One of the striking phenomena one can't help but notice in elementary Euclidean geometry is how easy it appears to be to define triples of lines in a triangle which meet in a point. Now for each ...
- 30.8k
3
votes
1
answer
82
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How big can a triangle be, whose sides are the perpendiculars to the sides of a triangle from the vertices of its Morley triangle?
Given any triangle $\varDelta$, the perpendiculars from the vertices of its (primary) Morley triangle to their respective (nearest) side of $\varDelta$ intersect in a triangle $\varDelta'$, which is ...
- 2,437
4
votes
1
answer
146
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The outer Nagel points and unknown central circle
Na, Nb, Nc are the outer Nagel points. A'B'C' is the contact triangle. I claim that lines A'B', A'C', B'C' always cut the sides of the triangle NaNbNc at six points corresponding to an unknown circle.
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- 445
4
votes
0
answers
366
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Two triangles have the same centroid theorem
Let $\triangle ABC$ and $\triangle A'B'C'$ be two triangles. The line through $A$ and perpendicular to $AA'$ meets the line through $B'$ and perpendicular to $BB'$ at $A_b$; The line through $A$ and ...
- 1,478
25
votes
6
answers
2k
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Are there infinitely many "generalized triangle vertices"?
Briefly, I'd like to know whether there are infinitely many "generalized triangle centers" which - like the orthocenter - are indistinguishable from a vertex of the original triangle. This ...
- 20.6k
7
votes
3
answers
399
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Maximizing the area of a region involving triangles
I thought of a question while making up an exercise sheet for high school students, and posted it on MathStackExchange but did not receive an answer (the original post is here), so I thought perhaps ...
- 26k
1
vote
0
answers
90
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Pseudo-Droz-Farny circles
I would like to present a construction of 2 circles. These 2 circles are somewhat similar in appearance to the well known Droz-Farny circles that can be drawn for every isogonal conjugate pairs of ...
- 445
15
votes
1
answer
14k
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The 4th vertex of a triangle?
I was immensely surprised and amused by the idea of the fourth side of a triangle that was introduced by B.F.Sherman in 1993. 'Sherman's Fourth Side of a Triangle' by Paul Yiu is available here. ...
- 445
3
votes
1
answer
353
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Constructing an equilateral triangle using Tarski's axioms for geometry
In Euclid's first geometry proposition, he constructs an equilateral triangle given an arbitrary line segment. I was wondering if it was possible to prove this straight from Tarski's axioms for ...
4
votes
2
answers
210
views
Six conelliptic points
Can you prove the following proposition:
Proposition. Given an arbitrary triangle $\triangle ABC$. Let $D,E,F$ be the points on the sides $AB$,$BC$ and $AC$ respectively , such that $\frac{AB}{DA}=\...
- 2,713
2
votes
1
answer
794
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Three circles meet at a point [closed]
I am looking for the proof of the following proposition:
Proposition. Let $\triangle ABC$ be an arbitrary triangle with circumcenter $O$. Let $A',B',C'$ be a reflection points of the points $A,B,C$ ...
- 2,713