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Results tagged with triangles
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user 13093
0
votes
Three circles intersecting at one point
It is straightforward to see that $A',B',C'$ are reflections of the circumcenter $O$ with respect to $BC, CA,AB$. Therefore, the center of $(AOA')$ is just the intersection of the mediatrix of $OA$ wi …
- 2,222
1
vote
Intersection point of three circles
Switching the roles of $ABC$ and $A'B'C'$, consider the circles passing through the vertices of a triangle $A,B,C$, midpoints of the opposite sides $A',B',C'$ and the circumcenter $O$.
It is straightf …
- 2,222
0
votes
A generalization of Napoleon's theorem
Notice that triangles $ACD, AEB, FCB$ are similar. Working out the ratios of the sides and the angles one can see that:
Triangles $AI_1I_3$ and $ACE$ are similar. … Triangles $BI_1I_2$ and $BEC$ are similar. Rotating $I_1I_2$ around $B$ with angle $\frac{1}{2}\angle FBC$ makes $I_1 I_2$ parallel to $CE$. …
- 2,222
2
votes
0
answers
409
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Important lines in triangle - reverse problem
It is known that if three numbers $x,y,z$ are the lengths of the edges of some triangle, then there exists a triangle with medians of length $x,y,z$. Also, if $x,y,z>0$ (no condition imposed) there ex …
- 4,713