Found problems: 4
In a convex quadrilateral $ABCD$ sides $AB$ and $DC$ are both divided into $m$ equal parts by points $A, S_1 , S_2 , \ldots , S_{m-1} ,B$ and $D,T_1, T_2, \ldots , T_{m-1},C,$ respectively (in this order).
Similarly, sides $BC$ and $AD$ are divided into $n$ equal parts by points $B,U_1,U_2, \ldots, U_{n-1},C$ and $A,V_1,V_2, \ldots,V_{n-1}, D$. Prove that for $1 \leq i \leq m-1$ each of the segments $S_i T_i$ is divided by the segments $U_j V_j$ ($1\leq j \leq n-1$) into $n$ equal parts
Is there a convex polygon with each side equal to some diagonal, and each diagonal equal to some side?
In the $\triangle ABC $, points $ M, I, H $ are feet, respectively, of the median, bisector and height, drawn from $ A $. It is known that $ BC = 2 $, $ MI = 2-\sqrt {3} $ and $ AB > AC $.
a) Prove that $ I$ lies between $ M $ and $ H $.
b) Calculate $ AB ^ 2-AC ^ 2 $.
c) Determine $ \dfrac {AB} {AC} $.
d) Find the measure of all the sides and angles of the triangle.
One triangle lies inside another. Prove that at least one of the two smallest sides (out of six) is the side of the inner triangle.