The lengths of three sides of a right triangle are all integer numbers.
a) Show that one of them is divisible by \(3\).
b*) Show that one of them is divisible by \(5\).
A segment \(AB\) is a base of an isosceles triangle \(ABC\). A line perpendicular to the segment \(AC\) was drawn through point \(A\) – this line crosses an extension of the segment \(BC\) at point \(D\). There is also a point \(E\) somewhere, such that angles \(\angle ECB\) and \(\angle EBA\) are both right. Point \(F\) is on the extension of the segment \(AB\), such that \(B\) is between \(A\) and \(F\). We also know that \(BF = AD\). Show that \(ED =EF\).
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 \(ABC\).
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.
A Cartesian plane is coloured in in two colours. Prove that there will be two points on the plane that are a distance of 1 apart and are the same colour.
All the points on the edge of a circle are coloured in two different colours at random. Prove that there will be an equilateral triangle with vertices of the same colour inside the circle – the vertices are points on the circumference of the circle.
Prove that no straight line can cross all three sides of a triangle, at points away from the vertices.
The angles of a triangle are in the ratio \(2: 3: 4\). Find the ratio of the outer angles of the triangle.
One angle of a triangle is equal to the sum of its other two angles. Prove that the triangle is right-angled.
In the acute-angled triangle \(ABC\), the heights \(AA_1\) and \(BB_1\) are drawn. Prove that \(A_1C \times BC = B_1C \times AC\).