In acute $\triangle ABC$, show $DE+DF \leq BC$, where $D$, $E$, $F$ are the feet of the altitudes from $A$, $B$, $C$, respectively.
108723
BM01 2011/12 Question 6 Geometry Problem
27272
Triangle Inequality with Complex Numbers
112969
Show that $ab + bc + ac \ge a\sqrt{bc} + b\sqrt{ac} + c\sqrt{ab}$
63958
how to prove this inequality $(ab+bc+ac)^2 ≥ 3abc(a+b+c)$
77576
Using the triangle inequalities to prove a bound in the complex plane
91325
Finding acute triangle probability
115929
In a triangle $\angle A = 2\angle B$ iff $a^2 = b(b+c)$
97473
Prove that: area of triangle $S_{ABC}=p \cdot R$, where p is half perimeter of triangle $DEF$
25870
Prove that $0 \leq ab + ac + bc - abc \leq 2.$
104217
Alternative triangle inequality proof
37233
Find the condition in which $\angle ADB = 3\angle BAC$
62826
The concurrence of angle bisector, median, and altitude in an acute triangle
6613
AM-GM inequality: $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{d} + \frac{d^2}{a} \geq a + b + c + d$
151154
Prove the inequality $\sqrt\frac{a}{a+8} + \sqrt\frac{b}{b+8} +\sqrt\frac{c}{c+8} \geq 1$ with the constraint $abc=1$
140299
Inequality based on triangle: $\frac{3}{2}\le\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$
14951
Other Idea to show an inequality $\dfrac{1}{\sqrt 1}+\dfrac{1}{\sqrt 2}+\dfrac{1}{\sqrt 3}+\cdots+\dfrac{1}{\sqrt n}\geq \sqrt n$
96065
Question about sines of angles in an acute triangle