Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

1996 OIM Problems/Problem 2

Revision as of 15:02, 13 December 2023 by Tomasdiaz (talk | contribs) (Created page with "== Problem == Let <math>M</math> be the midpoint of the median <math>AD</math> of triangle <math>ABC</math> (<math>D</math> belongs to side <math>BC</math>). The line <math>BM...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Problem

Let $M$ be the midpoint of the median $AD$ of triangle $ABC$ ($D$ belongs to side $BC$). The line $BM$ cuts the side $AC$ at the point $N$. Prove that $AB$ is tangent to the circumcircle of the triangle $NBC$ if, and only if, the following equality is verified

\[\frac{\overline{BM}}{\overline{MN}} = \frac{(\overline{BC})^2}{(\overline{BN})^2}\]

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Solution

This problem needs a solution. If you have a solution for it, please help us out by adding it.

See also

https://www.oma.org.ar/enunciados/ibe11.htm

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1996_OIM_Problems/Problem_2&oldid=207081"