Chapter 10.8, Problem 4E

To determine

Draw the nine-point circle for a right triangle. Also, check the number of points which are at the vertex of the right angle.

Answer to Problem 4E

The figure is:

  

There are 4 points which are present on the vertex of the right angle.

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.

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

Here, $H$ is the orthocenter which is same as vertex $A$ . Also, $X,T\text{and}R$ are same as on vertex $A$ .

Hence, the 4 points are on the vertex of the right angle.