Tasks

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TASKS

1. Find all triangles with integer sides whose area is expressed by the same number as perimeter.
2. There are two non-parallel segments $AB$ and $CD$ on the data plane. Construct a point $P$ such that the triangles $PAB$ and $PCD$ are similar, and the angles $APB$ and $CPD$ are equal.
3. Given a triangle $ABC$. Construct a point $M$ such that if $A_1, B_1, C_1$ is the intersection point of the lines $BC$ and $AM, CA$ and $BM, AB$ and $CM$, then $M$ is the center of gravity of the triangle $A_1B_1C_1$. Generalize the problem to the case when $M$ is the center of gravity of a system of given masses $m_A, m_B, m_C$ placed at points $A_1, B_1, C_1$.
4. Given a triangle $ABC$. At its heights the segments $AA_1, BB_1, CC_1$ are laid, having corresponding lengths $l$, $m$, $n$. Find the plane of triangle $A_1B_1C_1$.
5. Prove that for any triangle on the right the left unequal $p^2 - bc > S\sqrt{3}$ ($a, b, c$ is the side length, $p$ is the half-perimeter, $S$ is the area of ​​the triangle).
6. Given a triangle $ABC$. Let us draw a circle with center $A$ and radius equal to height $AD$, and a straight line through the points of intersection of this circle with sides $AB$ and $AC$. We will perform a similar construction for the other two vertices of the triangle. Prove that the resulting lines intersect at points $A_1, B_1, C_1$ such that

1. points $A_1, B_1, C_1$ lie on the bissertrices of angles $A, B, C$ of triangle $ABC$;
2. triangle $A_1B_1C_1$ is similar to a triangle whose vertices are the point of tangency of the side of triangle $ABC$ with the incircle;
3. the center of the circle inscribed at $ABC$ is the point of intersection of the altitudes of triangle $A_1B_1C_1$.