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Problem #PRU-108604

Problems Geometry Plane geometry Triangles Relationship of side lengths and angles of a triangle. Solving triangles. Ceva's theorem and Menelaus's theorem Calculus Integral Existence of a definite integral Vectors Law of polygon of vectors Similar triangles

13–14 3.0

On the sides \(AB\), \(BC\) and \(AC\) of the triangle \(ABC\) points \(P\), \(M\) and \(K\) are chosen so that the segments \(AM\), \(BK\) and \(CP\) intersect at one point and \[\vec{AM} + \vec{BK}+\vec{CP} = 0\] Prove that \(P\), \(M\) and \(K\) are the midpoints of the sides of the triangle \(ABC\).

Problem #PRU-116087

Problems Geometry Plane geometry Triangles Similar triangles Circles Tangent lines and tangent circles Tangent lines to circles

15–16 4.0

Three circles are constructed on a triangle, with the medians of the triangle forming the diameters of the circles. It is known that each pair of circles intersects. Let \(C_{1}\) be the point of intersection, further from the vertex \(C\), of the circles constructed from the medians \(AM_{1}\) and \(BM_{2}\). Points \(A_{1}\) and \(B_{1}\) are defined similarly. Prove that the lines \(AA_{1}\), \(BB_{1}\) and \(CC_{1}\) intersect at the same point.

Problem #PRU-56451

Problems Geometry Plane geometry Triangles Similar triangles The triangle generated by bases of two altitudes and a vertex

13–14 2.0

In the acute-angled triangle \(ABC\), the heights \(AA_1\) and \(BB_1\) are drawn. Prove that \(A_1C \times BC = B_1C \times AC\).

Problem #PRU-56508

Problems Geometry Plane geometry Triangles Similar triangles The triangle generated by bases of two altitudes and a vertex

12–14 2.0

Let \(AA_1\) and \(BB_1\) be the heights of the triangle \(ABC\). Prove that the triangles \(A_1B_1C\) and \(ABC\) are similar.

Problem #PRU-65204

Problems Geometry Plane geometry Triangles Similar triangles Auxiliary similar triangles Methods Extremal principle Extremal principle (other) Pigeonhole principle Pigeonhole principle (area and volume)

14–16 4.0

A unit square is divided into \(n\) triangles. Prove that one of the triangles can be used to completely cover a square with side length \(\frac{1}{n}\).

Problem #WSP-000163

Problems Geometry Plane geometry Triangles Similar triangles

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.

Problem #WSP-000164

Problems Geometry Plane geometry Triangles Similar triangles

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

Problem #WSP-000189

Problems Geometry Plane geometry Triangles Similar triangles

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

Problem #WSP-000190

Problems Geometry Plane geometry Triangles Similar triangles

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

Problem #WSP-000191

Problems Geometry Plane geometry Triangles Similar triangles

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 \(BD\)?