Triangles

Diagonals $AC$ and $BD$ of a trapezium $ABCD$ with $AB \parallel DC$ intersect each other at the point $O$. Using a similarity criterion for triangles, show that $\frac{AO}{OC} = \frac{BO}{OD}$.

State and prove the Pythagoras Theorem (Theorem 6.8 of NCERT Class 10 Mathematics).

State and prove the Basic Proportionality Theorem (Thales's Theorem). Also, in triangle ABC, DE is parallel to BC intersecting AB at D and AC at E. If AD = 1.5 cm, DB = 3 cm, and AE = 1 cm, find the length of EC.