Let \(ABC\) and \(DEF\) be two triangles such that \(\angle ACB = \angle DFE\) and \(\frac{DF}{AC} = \frac{EF}{BC}\). Prove that triangles \(ABC\) and \(DEF\) are similar.
Let \(AA_1\) and \(BB_1\) be the medians of the triangle \(ABC\). Prove that triangles \(A_1B_1C\) and \(BAC\) are similar. What is the similarity coefficient?
Let \(AD\) and \(BE\) be the heights of the triangle \(ABC\), which intersect at the point \(F\). Prove that the triangles \(AFE\) and \(BFD\) are similar.
Let \(AD\) and \(BE\) be the heights of the triangle \(ABC\). Prove that triangles \(DEC\) and \(ABC\) are similar.
Let \(CB\) and \(CD\) be tangents to the circle with the centre \(A\), let \(E\) be the point of intersection of the line \(AC\) with the circle. Draw \(FG\) as the segment of a tangent drawn through the point \(E\) between the lines \(CB\) and \(CD\). Find \(FG\) if the radius of the circle is \(15\) and \(AC = 39\).
In the triangle \(ABC\) with a right angle \(\angle ACB\), \(CD\) is the height and \(CE\) is the bisector. Draw the bisectors \(DF\) and \(DG\) of the triangles \(BDC\) and \(ADC\). Prove that \(CFEG\) is a square.