# The bisectors of the angles B and C of the trapezoid ABCD intersect at the point O lying on the side AD

May 31, 2020 | Education

| **The bisectors of the angles B and C of the trapezoid ABCD intersect at the point O lying on the side AD. Prove that the point O is equidistant from the lines AB, BC and CD.**

The distance from point O to lines is the length of the perpendicular drawn from the point to the line. In other words, one must prove that ON = OM = OK.

Consider the triangle NBO.

sin∠NBO = ON / OB (by definition of the sine).

ON = OB * sin∠NBO

Consider the triangle BMO.

sin∠OBM = OM / OB (by definition of the sine).

OM = OB * sin∠OBM

∠NBO = ∠OBM (since OB is a bisector).

Therefore, OM = OB * sin∠OBM = OB * sin∠NBO = ON

It is similarly proved that OK = OM.

Those. ON = OM = OK.

Remember: The process of learning a person lasts a lifetime. The value of the same knowledge for different people may be different, it is determined by their individual characteristics and needs. Therefore, knowledge is always needed at any age and position.