Problems

The triangle \(ABC\) is inscribed into the circle with centre \(E\), the line \(AD\) is perpendicular to \(BC\). Prove that the angles \(\angle BAD\) and \(\angle CAE\) are equal.

On the diagram below \(BC\) is the tangent line to a circle with the centre \(A\), and it is known that the angle \(\angle ABC = 90^{\circ}\). Prove that the angles \(\angle DEB\) and \(\angle DBC\) are equal.

The triangle \(BCD\) is inscribed in a circle with the centre \(A\). The point \(E\) is chosen as the midpoint of the arc \(CD\) which does not contain \(B\), the point \(F\) is the centre of the circle inscribed into \(BCD\). Prove that \(EC = EF = ED\).

Let \(\triangle ABC\) and \(\triangle DEF\) be triangles such that the following angles are equal: \(\angle ABC = \angle DEF\) and \(\angle ACB = \angle DFE\). Prove that \(\triangle ABC\) and \(\triangle DEF\) are similar triangles.

The medians \(AD\) and \(BE\) of the triangle \(\triangle ABC\) intersect at the point \(F\). Prove that \(\triangle AFB\) and \(\triangle DFE\) are similar. What is their similarity coefficient?

In a triangle \(\triangle ABC\), the angle \(\angle B = 90^{\circ}\) . The altitude from point \(B\) intersects \(AC\) at \(D\). We know the lengths \(|AD|=9\) and \(|CD|=25\). What is the length \(|BD|\)?

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.