Chapter 10.8, Problem 3E

To determine

Draw the nine-point circle for an equilateral triangle.

Answer to Problem 3E

The figure is:

  

Explanation of Solution

Given information:

The circle has nine-point.

Calculation:

Consider a triangle $ABC$ .

The nine-point circle is:

  

  $L,M,\text{and}N$ are the midpoints of $AB,BC,\text{and}CA$ and $R,S,\text{and}T$ are the points on $AB,BC,\text{and}CA$ respectively.

In this triangle, the midpoints and the altitudes meet at the same point.

Thus, $T\text{and}N$ , $M\text{and}S$ and $L\text{and}R$ are same points.

Here, $H$ is the orthocenter and $O$ is the center of the circle.

  $X,Y,\text{and}Z$ are the midpoints of $HA,HB,\text{and}HC$ respectively.

Hence, the required construction.