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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-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. This is usually known as the “side-angle-side" criterion for congruence.

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 \(BD\)?

Problem #WSP-000192

Problems Geometry Plane geometry Triangles Similar triangles

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.

Problem #WSP-000193

Problems Geometry Plane geometry Triangles Similar triangles

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?

Problem #WSP-000194

Problems Geometry Plane geometry Triangles Similar triangles

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.