( Answered) Modern Geometry

( Answered) Modern Geometry

( Answered) Modern Geometry

Provide answers for the following questions giving a reasoning behind each one, illustrations are provided
for the first and second problems for a better understanding.

1. For the following illustration, prove the the ABC angle is equal to (1/2)*AOC – (1/2)*DOE, in other
words, that the ABC angle is half of the “value” of the AOC minus half of the “value” of the DOE. O is the
center of the circumference.

2. In a town located next to a river, whose border is completely straight, there is a fire in a place marked
as A. Near the border of the river, there is the house of the firemen marked B; in order to put out the fire, the firemen must walk to the edge of the river to fill a bucket with water and run over to point B to throw it over the fire. Which point R (or points) on the edge of the river would make the trajectory as small as possible (i.e. what point on the river would minimize the ARB distance)?

3. Let ABC be a triangle such that angle B is a right angle. Let L be the midway point of the side AC, let X

be the point on which the bisector of the right angle B crosses the side AC and let D be the point on which

the height drawn from B cuts the side AC. Show that the angle DBX is equal to the angle XBL.