Questions tagged [circumcircle]
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5 questions
3 votes
2 answers
333 views
What is the nature of a triangle for which the square of the diameter of its circumcircle is equal to the sum of the squares of two of its sides?
Fifty years ago , when I was in college, our teacher , Mrs Marie -Jo , gave us this homework assignment for the holidays : " Find the two types of triangles such that the square of the diameter ...
- 1,898
6 votes
3 answers
632 views
Find a synthetic proof to an old problem .
I found this problem in a French paper translated from Arabic in 1927 by an author named Al Bayrouni . I wonder if it can be found in one of Archimedes' works . Here is the statement : ABC is a ...
- 1,898
2 votes
0 answers
139 views
How can we calculate the radius of the circumcircle of triangle ABC under these conditions?
We know that if H is the orthocenter of triangle ABC , then the circumcircles of triangles HAB , HAC and HBC all have the same radius R : the radius of the circumcircle of ABC . So , I asked myself ...
- 1,898
3 votes
3 answers
161 views
Proving the tangency of two circles
I need help regarding this question is from the Philippine Math Olympiad 2025: Let $ABC$ be a triangle with incenter $I$, and let $D$ be a point on side $BC$. Points $X$ and $Y$ are chosen on lines $...
- 307
3 votes
1 answer
205 views
A symmetric characterization of triangles with two angles differing by $90^\circ$
For several weeks, I have been wondering whether it is possible to find a symmetric, neutral formula characterizing triangles whose two angles differ by $90^\circ$. By neutral I mean a condition that ...
- 4,501