Chapter 10.3, Problem 2EX

To determine

the relationship between $\overline{AE}$ and $\overline{AB}$ , $\overline{BF}$ and $\overline{BC}$ , $\overline{CG}$ and $\overline{CA}$ .

Answer to Problem 2EX

The lines drawn perpendicular to all sides of a triangle from the centre of the circumscribed circle, divide the sides into the ratio of $1:1$ .

Explanation of Solution

Given:

In triangle ABC from centre of the circumscribed circle, three lines are drawn perpendicular to the three sides of this triangle.

Calculation:

From the figure, we can write:

Lines drawn perpendicular to all sides of a triangle from the centre of the circumscribed circle, divide the sides into the ratio of $1:1$ . Thus, we have:

  $\begin{array}{l}\overline{AE}=\frac{1}{2}\overline{AB},\\ \overline{BF}=\frac{1}{2}\overline{BC},\text{ and}\\ \overline{CG}=\frac{1}{2}\overline{CA}.\end{array}$