If $A$ and $a$ are the radii of a triangle's circum- and in-circles, and $b$ the distance between the circles' centers, prove $A^2-b^2=2 A a$
111042
A geometry problem from high school Olympiad selection round
153313
Prove that for all integers $ a $ and $ b $, if $ a $ divides $ b $, then $ a^{2} $ divides $ b^{2} $.
148147
Three circles having centres on the three sides of a triangle
37808
Show that $\gcd(a^2, b^2) = \gcd(a,b)^2$
45702
For any triangle with sides $a$, $b$, $c$, show that $a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge 0$
64497
prove that $\tan(\alpha+\beta) = 2ac/(a^2-c^2)$
11338
How to find the circumcenter of a triangle and the length of the corresponding radius of the circle in $\mathbb{R}^3$
27954
Prove triangle inequality using the properties of absolute value
118574
prove that $24 \mid a(a^2-1)$
75679
How to prove that $2 \arctan\sqrt{x} = \arcsin \frac{x-1}{x+1} + \frac{\pi}{2}$
127891
If $ a^{2} + b^{2} = c^{2}$ and $c$ is even, prove that $a$ and $b$ are both even.
87111
Prove $\sqrt{2}$ is between $\dfrac{a}{b}$ and $\dfrac{a+2b}{a+b}$
95136
Finding center and radius of circumscribed triangle
44119
Prove that $\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$
118501
Proving the length of a circle's arc is proportional to the size of the angle
62126
Constructing a triangle given 2 sides and the inscribed circle's radius
135365
If $x, y, z$ are the side lengths of a triangle, prove that $x^2 + y^2 + z^2 < 2(xy + yz + xz)$