Problems
Recall that a line is tangent to a circle if they have only one point of intersection, a circle is called inscribed in a polygon if it is tangent to every side as a segment of that polygon.
In the triangle \(EFG\) the line \(EH\) is the median. Two circles with centres \(A\) and \(C\) are inscribed into triangles \(EFH\) and \(EGH\) respectively, they are tangent to the median \(EH\) at the points \(B\) and \(D\). Find the length of \(BD\) if \(EF-EG=2\).
Is it possible to cover a \((4n+2) \times (4n+2)\) board with the \(L\)-tetraminos without overlapping for any \(n\)? The pieces can be flipped and turned.
On the diagram each number denotes the area of a rectangle it is written into. What are the areas of the other rectangles?
Line \(AB\) is parallel to line \(CD\) and line \(AD\) is parallel to line \(BE\). Show that triangles \(ADE\) and \(ABC\) have equal areas.
A quadrilateral \(ABCD\) is given. Points \(K\) and \(L\) belong to the side \(AB\) and \(AK=KL=LB\) and points \(N\) and \(M\) belong to the side \(CD\) and \(CM=MN=ND\). Show that the area of the quadrilateral \(KLMN\) is \(\frac13\) of the area of the quadrilateral \(ABCD\).
A quadrilateral \(ABCD\) is given. Point \(M\) is a midpoint of \(AB\) and point \(N\) is a midpoint of \(CD\). Point \(P\) is where segments \(AN\) and \(DM\) meet, point \(Q\) is where segments \(MC\) and \(NB\) meet. Show that the sum of the areas of triangles \(APD\) and \(BCQ\) is equal to the area of the quadrilateral \(MQNP\).
A square was cut with two parallel lines. The perpendicular distance between these two lines is \(6\)cm. One of them goes through the top right corner and the other through the bottom left corner. The three regions obtained this way, two triangles and a parallelogram, have equal areas. What is the area of the square?
The area of the triangle \(\triangle AEC\) is \(4\), the area of the triangle \(\triangle BCE\) is \(9\) and the area of the triangle \(\triangle ABC\) is \(21\). What is the area of the triangle \(\triangle ADE\)?
A series of squares are connected by touching vertices. Some triangles were drawn outside and inside of the chain. Show that the total green area is the same as the total red area.