In triangle $ABC,$ we have $\angle B = 2\angle A.$ Prove that $b^2 = a(a+c).$
115929
In a triangle $\angle A = 2\angle B$ iff $a^2 = b(b+c)$
87206
Prove that a triangle can be inscribed in the hyperbola $xy=c^2$ whose sides touch the parabola $y^2=4ax$.
50296
Prove $(-a+b+c)(a-b+c)(a+b-c) \leq abc$, where $a, b$ and $c$ are positive real numbers
106564
Suppose $A,B,C$ are sets. Prove that $A\mathbin\triangle B\subseteq C \iff A\cup C=B\cup C$.
42545
If $ a,b,c\in \left(0,\frac{\pi}{2}\right)\;,$ Then prove that $\frac{\sin (a+b+c)}{\sin a+\sin b+\sin c}<1$
2019
Prove $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$
84422
$AB=BA$, $AC=CA$ we need to show $BC=CB$
145783
How can I prove this inequality in a triangle: $\frac{1}{(b+c-a)} +\frac{1}{(c+a-b)} +\frac{1}{(a+b-c)} \gt \frac{1}{a} +\frac{1}{b} +\frac{1}{c}$?
49548
If $(a + ib)^3 = 8$, prove that $a^2 + b^2 = 4$
139082
If $a,b,c>0$ and $a^3+b^3=c^3$ then prove that $a^2+b^2-c^2>6(c-a)(c-b)$.
21726
Show that if $(b,c)=1$, then $(a,bc)=(a,b)(a,c)$.
90902
Prove $\frac{a+b+c}{abc} \leq \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}$.
4333
Prove that if $a+b+c+d=4$, then $(a^2+3)(b^2+3)(c^2+3)(d^2+3)\geq256$
100571
Prove $(1-a)(1-b)(1-c)(1-d)\geq abcd$ if $a^2+b^2+c^2+d^2=1$
124425
A geometry question: Let $ABCD$ be a convex quadrilateral. Prove that AD is parallel to BC.
130470
Find the angle ADE of the given triangle.
43183
basic geometry angle problem