### 1967 IMO SL #4

Suppose medians $m_a$ and $m_b$ of a triangle are orthogonal. Prove that

a) Using medians of that triangle it is possible to construct a rectangular triangle.

b) The following inequality:\[5(a^2+b^2-c^2) \geq 8ab,\]is valid, where $a,b$ and $c$ are side length of the given triangle.

a) Let $ABCD$ be a parallelogram and $K,L$ the midpoints of segments $BC, CD$. The sides of $\triangle{AKL}$ are equal and parallel to the medians of triangle $ABC$. This proves the desired. $\square$

b) Using the formulas $4m_a^2=2b^2+2c^2-a^2$, etc., we can see that $$m_a^2+m_b^2=m_c^2 \implies a^2+b^2=5c^2.$$

Then, we have $$5(a^2+b^2-c^2)=4(a^2+b^2) \geq 8ab,$$ as desired. $\square$

nice solution

ReplyDeletepart a is very elegant

ReplyDeletegreat job

hi

ReplyDeleteyour part a was rlly elegant as @anonymous (comment#2) said

waiting for the next blog post :)

yeah nice sol

ReplyDeleteand also thanks for making this blog :D

i could get a decent score on the AMC 10

but im literally failing aime mocks (like actually i get 5s and 6s)

i prolly wont qual for jmo this year

but if i keep reading ur blog then i might next year :)