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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 $A B$ , $B C$ and $A C$ of the triangle $A B C$ points $P$ , $M$ and $K$ are chosen so that the segments $A M$ , $B K$ and $C P$ intersect at one point and $\vec{A M} + \vec{B K} + \vec{C P} = 0$ Prove that $P$ , $M$ and $K$ are the midpoints of the sides of the triangle $A B C$ .

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 $A M_{1}$ and $B M_{2}$ . Points $A_{1}$ and $B_{1}$ are defined similarly. Prove that the lines $A A_{1}$ , $B B_{1}$ and $C C_{1}$ 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 $A B C$ , the heights $A A_{1}$ and $B B_{1}$ are drawn. Prove that $A_{1} C \times B C = B_{1} C \times A C$ .

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 $A A_{1}$ and $B B_{1}$ be the heights of the triangle $A B C$ . Prove that the triangles $A_{1} B_{1} C$ and $A B C$ 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 $A B C$ and $A_{1} B_{1} C_{1}$ be two triangles with the following properties: $A B = A_{1} B_{1}$ , $A C = A_{1} C_{1}$ , and angles $∠ B A C = ∠ B_{1} A_{1} C_{1}$ . Then the triangles $A B C$ and $A_{1} B_{1} C_{1}$ 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 $△ A B C$ the sides $A C$ and $B C$ are equal. Prove that the angles $∠ C A B$ and $∠ C B A$ are equal.

Problem #WSP-000189

Problems Geometry Plane geometry Triangles Similar triangles

Let $A B C$ and $D E F$ be such triangles that angles $∠ A B C = ∠ D E F$ , $∠ A C B = ∠ D F E$ . Prove that the triangles $A B C$ and $D E F$ are similar.

Problem #WSP-000190

Problems Geometry Plane geometry Triangles Similar triangles

The medians $A D$ and $B E$ of the triangle $A B C$ intersect at the point $F$ . Prove that the triangles $A F B$ and $D F E$ are similar. What is their similarity coefficient?

Problem #WSP-000191

Problems Geometry Plane geometry Triangles Similar triangles

In a triangle $△ A B C$ , the angle $∠ B = 90^{\circ}$ . The altitude from point $B$ intersects $A C$ at $D$ . We know the lengths $A D = 9$ and $C D = 25$ . What is the length $B D$ ?