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Results tagged with euclidean-geometry
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user 88804
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these.
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 …
- 2,222
12
votes
2
answers
958
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 $AC$ , respectiv …
- 2,222
2
votes
2
answers
536
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 …
- 2,222
6
votes
4
answers
570
views
Necessary and sufficient condition for quadrilateral to be cyclic
Can you provide a proof for the following proposition:
Proposition. Given any quadrilateral $ABCD$. Let $P,Q,R,S$ be nine-point centers of triangles $\triangle ABD$,$\triangle ABC$,$\triangle BCD$ an …
- 1,986
2
votes
1
answer
798
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 $A,B,C$ w …
CommunityBot
- 1
3
votes
2
answers
270
views
Four concyclic points inside bicentric quadrilateral
Can you provide a proof for the following proposition:
Proposition. Let quadrilateral $ABCD$ be inscribed into a circle with center $O$ and circumscribed around a circle with center $I$. Let $X$ be a …
- 680
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 $J_ …
- 445
3
votes
1
answer
158
views
The product of the lengths of two line segments that belong to Newton line [closed]
I am looking for the proof of the following claim:
Consider a family of bicentric quadrilaterals with the same inradius length and the same distance between incenter and circumcenter. Denote by $P$ a …
1
vote
1
answer
104
views
Collinearity in tangential pentagon [closed]
I am looking for a proof of the following claim:
Given tangential pentagon. Touching point of the incircle and the side of the pentagon,the vertex opposite to that side and the intersection point of …
- 5,901
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}=\ …
- 109k
2
votes
0
answers
82
views
Principal diagonals of octagon meet in a single point
Can you provide a proof for the following claim:
Claim. Given octagon circumscribed about an ellipse. If the vertices of the octagon lie on another ellipse then its principal diagonals meet in a sing …
- 2,661
1
vote
1
answer
84
views
Equal products of triangle areas
Can you prove the following claim:
Claim. Given hexagon circumscribed about an ellipse. Let $A_1,A_2,A_3,A_4,A_5,A_6$ be the vertices of the hexagon and let $B$ be the intersection point of its princ …
- 5,901
1
vote
1
answer
155
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 …
- 266
3
votes
1
answer
103
views
Equal sums of line segments
I would like to see a proof of the following
Claim. Let $A_1,A_2,A_3,A_4,A_5$ be vertices of bicentric pentagon. Let $B_1$ be the intersection point of $A_1A_3$ and $A_2A_5$, $B_2$ the intersection p …
- 28.9k
2
votes
1
answer
182
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 s …
- 1,848