All Questions
Tagged with elementary-proofs triangles
15 questions
35
votes
17
answers
6k
views
Which theorems have Pythagoras' Theorem as a special case?
Loomis famously wrote hundreds of proofs of Pythagoras' Theorem (reference below), but these are all basically proofs "from below". Today on Twitter @panlepan mentioned Carnot's theorem ...
17
votes
2
answers
1k
views
Why are the medians of a triangle concurrent? In absolute geometry
This fact holds true in absolute geometry, and I would like to see an elementary synthetic proof not using the classification of absolute planes (Euclidean and hyperbolic planes) and specific models. ...
- 109k
15
votes
1
answer
17k
views
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
14
votes
4
answers
1k
views
Six points on an ellipse
Can you prove the following proposition:
Proposition. Let $\triangle ABC$ be an arbitrary triangle with centroid $G$. Let $D,E,F$ be the points on the sides $AC$,$AB$ and $BC$ respectively , such ...
- 2,661
12
votes
2
answers
969
views
Intersection point of three circles
Can you provide a proof for the following proposition:
Proposition. Let $\triangle ABC$ be an arbitrary triangle with orthocenter $H$. Let $D,E,F$ be a midpoints of the $AB$,$BC$ and $AC$ , ...
- 2,661
8
votes
4
answers
2k
views
Three circles intersecting at one point
Can you provide a proof for the following proposition:
Proposition. Let $\triangle ABC$ be an arbitrary triangle with nine-point center $N$ and circumcenter $O$. Let $A',B',C'$ be a reflection points ...
- 2,661
4
votes
1
answer
215
views
Point of concurrency [closed]
I am looking for the proof of the following claim:
Claim: Let $\triangle ABC$ be an arbitrary triangle, $D$ its nine-point center and $E,F,G$ are the nine-point centers of the triangles $\triangle ...
- 2,661
4
votes
2
answers
211
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}=\...
- 2,661
2
votes
1
answer
802
views
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 $A,B,C$ ...
- 2,661
2
votes
1
answer
202
views
The centroid, the first and second Napoleon points and $X(930)$ lie on a circle
Can you provide an elementary proof for the claim given below?
Preliminary definitions:
$X(110)=$ focus of Kiepert parabola.
$X(137)=X(110)$ of orthic triangle .
$X(930)=$ anticomplement of $X(137)$ .
...
- 2,661
2
votes
2
answers
537
views
A generalization of Napoleon's theorem
Can you provide a proof for the following proposition?
Proposition. Given an arbitrary $\triangle ABC$. The $\triangle AEB$, $\triangle BFC$ and $\triangle CDA$ are constructed on the sides of the $...
- 2,661
2
votes
2
answers
247
views
Six concyclic points
Can you provide a proof for the following proposition:
Proposition. Let $\triangle ABC$ be an arbitrary triangle with excenters $J_A$,$J_B$ and $J_C$ . Let $G$ be the orthogonal projection of the $...
- 2,661
2
votes
1
answer
184
views
Four concyclic triangle centers
Can you prove the claim given below? Inspired by Lester's theorem I have formulated the following claim:
Claim. Given any scalene triangle $\triangle ABC$ . Let $D$ be the reflection of incenter in ...
- 2,661
1
vote
1
answer
159
views
Generalizing Bottema's theorem
Can you provide another proof for the claim given below?
Claim. In any triangle $\triangle ABC$ construct triangles $\triangle ACE$ and $\triangle BDC$ on sides $AC$ and $BC$ such that $\frac{AE}{AC}...
- 2,661
1
vote
0
answers
96
views
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