Problems

Let \(\triangle ABC\) and \(\triangle DEF\) be two triangles such that \(\angle ACB = \angle DFE\) and \(\frac{|DF|}{|AC|} = \frac{|EF|}{|BC|}\). Prove that \(\triangle ABC\) and \(\triangle DEF\) are similar.

Let \(AA_1\) and \(BB_1\) be the medians of the triangle \(\triangle ABC\). Prove that \(\triangle A_1B_1C\) and \(\triangle BAC\) are similar. What is the similarity coefficient?

Let \(AD\) and \(BE\) be the heights of the triangle \(\triangle ABC\), which intersect at the point \(F\). Prove that \(\triangle AFE\) and \(\triangle BFD\) are similar.

Let \(AD\) and \(BE\) be the heights of the triangle \(\triangle ABC\). Prove that \(\triangle DEC\) and \(\triangle ABC\) 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 the length \(|FG|\) if the radius of the circle is \(15\) and \(|AC| = 39\).

In triangle \(\triangle ABC\) with right angle \(\angle ACB=90^{\circ}\), \(CD\) 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.

The perimeter of the triangle \(\triangle ABC\) is \(10\). Let \(D,E,F\) be the midpoints of the segments \(AB,BC,AC\) respectively. What is the perimeter of \(\triangle DEF\)?

A circle with centre \(A\) has the point \(B\) on its circumference. A smaller circle is drawn inside this with \(AB\) as a diameter and \(C\) as its centre. A point \(D\) (not \(B\)!) is chosen on the circumference of the bigger circle, and the line \(BD\) is drawn. \(E\) is the point where the line \(BD\) intersects the smaller circle.

Show that \(|BE|=|DE|\).