Two circles have radii \(R_1\) and \(R_2\), and the distance between their centers is \(d\). Prove that these circles are orthogonal if and only if \(d^2 = R_1^2 + R_2^2\).
Let \(E\) and \(F\) be the midpoints of the sides \(BC\) and \(AD\) of the parallelogram \(ABCD\). Find the area of the quadrilateral formed by the lines \(AE, ED, BF\) and \(FC\), if it is known that the area \(ABCD\) is equal to \(S\).
Let \(G,F,H\) and \(I\) be the midpoints of the sides \(CD, DA, AB, BC\) of the square \(ABCD\), whose area is equal to \(S\). Find the area of the quadrilateral formed by the straight lines \(BG,DH,AF,CE\).
The diagonals of the quadrilateral \(ABCD\) intersect at the point \(O\). Prove that \(S_{AOB} = S_{COD}\) if and only if \(BC \parallel AD\).
a) Prove that if in the triangle the median coincides with the height then this triangle is an isosceles triangle.
b) Prove that if in a triangle the bisector coincides with the height then this triangle is an isosceles triangle.
Prove that the bisectors of a triangle intersect at one point.
A circle divides each side of a triangle into three equal parts. Prove that this triangle is regular.
Prove that the following inequalities hold for the Brockard angle \(\varphi\):
a) \(\varphi ^{3} \le (\alpha - \varphi) (\beta - \varphi) (\gamma - \varphi)\) ;
b) \(8 \varphi^{3} \le \alpha \beta \gamma\) (the Jiff inequality).
Prove that a convex quadrilateral \(ABCD\) can be drawn inside a circle if and only if \(\angle ABC + \angle CDA = 180^{\circ}\).
Prove that a convex quadrilateral \(ICEF\) can contain a circle if and only if \(IC+EH = CE+IF\).