Problems

Let \(ABC\) and \(A_1B_1C_1\) be two triangles with the following properties: \(AB = A_1B_1\), \(AC = A_1C_1\), and angles \(\angle BAC = \angle B_1A_1C_1\). Then the triangles \(ABC\) and \(A_1B_1C_1\) are congruent. This is usually known as the “side-angle-side" criterion for congruence.

In the triangle \(\triangle ABC\) the sides \(AC\) and \(BC\) are equal. Prove that the angles \(\angle CAB\) and \(\angle CBA\) are equal.

Let \(ABC\) be a triangle with given angles \(\angle BAC\) and \(\angle ABC\). What is the value of the angle \(\angle BCD\) in terms of \(\angle BAC\) and \(\angle ABC\)?

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.

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.