*2. Centers of Triangle -Generally, most students know that there is a center only in a circle. They don’t notice that there are centers, namely circumcenter, centroid called center of gravity, incenter and orthocenter in a triangle. Now let’s study how they are located in a triangle. **2.1 Circumcenter of a Triangle -Construct perpendicular bisectors of the segments of the triangle ABC on a piece of paper using a pencil and ruler. Label the point of intersection of the segment BC and the perpendicular bisector on it as D, the segment AC and the perpendicular bisector on it as E, and the segment AB and the perpendicular bisector on it as F. <center> %02-01.png </center> --(1) What happen to the perpendicular bisectors in your drawing? -@02a.htm#2.1.1, click here to investigate by computer or GC/html (2.1.1) --(2) What happens to them now? ---They intersect exactly at one point. This point is called the circumcenter of the triangle. Label it “O”. -Check the following questions (3) to (6) using the figure that you have had by clicking "by computer or GC/html" above. --(3) If you drag the vertex A to the other places to reshape the picture, what happens to the circumcenter O? --(4) If the triangle is acute (all angles smaller than a right angle), where is the circumcenter? --(5) If the triangle is obtuse (has an angle bigger than a right angle), where is it? --(6) If the triangle is right triangle, where is it? -Now you have learned that the perpendicular bisectors of a triangle always intersect at one point. In other words, all triangles have their circumcenter O. -So draw a circle taking a radius from point O to point A. --(7) Which points does the circle pass through? -@02a.htm#2.1.12, click here to investigate by computer or GC/html (2.1.2) --(8) Looking at the circumcircle, what can you say about the segments OA, OB, and OC? Why? Prove your finding. --(9) Do you notice that no matter how you move the point A to other places, the circumcenter is always on the perpendicular bisector of BC? --(10) If so, where is the circumcenter if you move the point B, or C? -Click "by computer or GC/html 2.3" below to check the answers of question (8) to question (10). -@02a.htm#2.1.3, click here to investigate by computer or GC/html (2.1.3) **2.2 Centroid or Center of Gravity of a Triangle -Construct medians of the segments of the triangle ABC on a piece of paper using a pencil and straightedge. <center> %02-03.png </center> --(1) What happens to the medians in your drawing? -@02a.htm#2.2.1, click here to investigate by computer or GC/html (2.2.1) --(2) What happens to them now? ---They intersect exactly at one point. This point is called the centroid or center of gravity of the triangle. Label it “G”. --(3) Check the centroid “G” by dragging any vertex of the triangle ABC. ---i. If the triangle is acute, where is the centroid? ---ii. If the triangle is obtuse, where is it? ---iii. If the triangle is right triangle, where is it? --(4) Compare the centroid with the circumcenter. What are the differences between them? --(5) Prove the three medians divide the triangle into six smaller triangles that all have the same area, even though they may have different shapes. (See the figure below.) <center> %01-11.png </center> --(6) Try this: ---i. Make any triangle about “9-18 inches” wide from cardboard. Make it as lopsided and irregular as you can. ---ii. Make a small hole near a vertex (any one you want) on the cardboard triangle. ---iii. Tie a string through the small hole. Let both ends of the string protrude. ---iv. Put a weight of some kind on one end of the string. ---v. Hold up the triangle by the end of the string with the weight. At the same time, there will be a line created by the string on the triangle by the weight end. (See the figure below.) ---vi. What can you say about the line on the triangle? Why? <center> %02-04.png </center> **2.3 Orthocenter of a Triangle -Construct three altitudes of the triangle ABC on a piece of paper using a pencil and straightedge. <center> %02-05.png </center> --(1) What happens to the altitudes in your drawing? -@02a.htm#2.3.1, click here to investigate by computer or GC/html (2.3.1) --(2) What happens to them now? ---They intersect exactly at one point. This point is called the orthocenter of the triangle. Label it “H”. -Drag a vertex of the triangle and observe the orthocenter. --(3) Where is the orthocenter if the triangle is acute? --(4) Where is it if the triangle is obtuse? (Extend the sides of the triangle if it is necessary.) --(5) Where is it if the triangle is right triangle? --(6) What are the differences between the orthocenter and the circumcenter? How about the orthocenter and the centroid? **2.4 Incenter of a Triangle -Construct angle bisectors of the triangle ABC on a piece of paper using a pencil and ruler. <center> %02-06.png </center> --(1) What happens to the angle bisectors in your drawing? -@02a.htm#2.4.1, click here to investigate by computer or GC/html (2.4.1) --(2) What happens to them now? ---They intersect exactly at one point. This point is called the incenter of the triangle. Label it “I”. -Drag a vertex of the triangle in order to observe the incenter. --(3) Where is the incenter if the triangle is acute? --(4) Where is it if the triangle is obtuse? --(5) Where is it if the triangle is right triangle? --(6) Compare the incenter with circumcenter, centroid, and orthocenter. What can you conclude from your comparison? **2.5 Further Investigations ***2.5.1 Euler Line -Construct a circumcenter (O), centroid (G), orthocenter (H) and incenter (I) in the same triangle using a pencil, compass and ruler as shown in figure below. And draw a line that passes through the circumcenter (O) and orthocenter (H) of the triangle. <center> %02-07.png </center> --(1)Which points does the line pass through in your construction? -@02a.htm#2.5.1.1, click here to investigate by computer or GC/html (2.5.1.1) -Drag any vertex of the triangle to observe your answer in question (1). ---The line always passes through the circumcenter, centroid and orthocenter of the triangle. This line is called the Euler line, named after Leonhard Euler (1707-1783). Euler showed in 1765 that the circumcenter (O), centroid (G) and orthocenter (H) of a triangle are collinear. This property is also true for another triangle center, nine-point center, although it had not been defined in Euler’s time. (See the figure below and nine-point center is explained in the following section.) <center> %02-08.png </center> ---The Euler line is the red line that passes through the circumcenter (green), centroid(orange), orthocenter(blue) and center of the nine-point circle(red). --(2) Drag any vertex of the triangle to be an isosceles triangle. What do you notice about the four centers of the triangle? --(3) Then, drag any vertex again to be an equilateral triangle. What happens to the four centers of the triangle? --(4) Where is the Euler line in the case of a right triangle? Explain your answer. --(5) How does the Euler line divide the triangle ABC when it is an isosceles triangle? -Measure the distance from the circumcenter (O) to the centroid (G) and that from the centroid (G) to orthocenter (H) on the Euler line. --(6) What is the ratio of OG and GH? ---No matter how the triangle’s shape is, the ratio of the distance between the circumcenter (O) and the centroid (G) to that between the centroid (G) and the orthocenter (H) is 1:2. (In figure below, OG:GH=1:2) <center> %02-09.png </center> ---In the figure above, if you drag the vertex A to anywhere, the values of OG and GH change but that of OG/GH does not change. -@02a.htm#2.5.1.2, click here to investigate by computer or GC/html (2.5.1.2) ***2.5.2 Nine-point Circle -In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are: ---The midpoint of each side of the triangle ---The foot of each altitude ---The midpoint of the line segment from each vertex of the triangle to the orthocenter (where the three altitudes meet; these line segments lie on their respective altitudes). (See the figure below.) <center> %02-10.png </center> --(1) How can you find the location of the nine-point center? Explain. --(2) Is the nine-point center always inside the triangle? -@02a.htm#2.5.2.1, click here to investigate by computer or GC/html (2.5.2.1) ***2.5.3 Incircle and Circumcircle -You have already learned the incenter of the triangle. You know that you can draw many circles with various sizes inside the triangle. --(1) Do you know which circle would be the largest among them? Check this with the following steps. ---i. Draw a circle as large as you can, but inside the triangle. ---ii. Measure and note the radius or diameter of your circle. ---iii. Again, draw another circle making the center be the incenter of the triangle. ---iv. Measure and note the radius or diameter of the circle. ---v. Compare the radii or diameters of both of circles. Which one is longer? (The longer radius circle is larger.) ---The circle in which the center is the incenter of the triangle is the largest circle inside the triangle. This circle is called the incircle or the inscribed circle of the triangle. It always fits inside the triangle. Each of the triangle’s three sides is tangent to the circle. (See the figure below.) <center> %02-11.png </center> -@02a.htm#2.5.3.1, click here to investigate by computer or GC/html (2.5.3.1) --(2) You know that you can draw many circles that surround a triangle. Which circle is the smallest among them? (In the figure below, the red and green circles are circles that surround the triangle ABC.) <center> %02-12.png </center> -@02a.htm#2.5.3.2, click here to investigate by computer or GC/html (2.5.3.2) --(3) You can infer that there are many triangles with sides that are tangent to a circle. In other words, there are many tangential triangles in a circle. Which triangle would be the smallest sum of the three sides among them? (In the figure below, ABC and DEF are the tangential triangles.) <center> %02-13.png </center> -@02a.htm#2.5.3.3, click here to investigate by computer or GC/html (2.5.3.3) --(4) You can draw many triangles inside a circle that its vertices are touching the circle. Which triangle would be the largest area? (See the figure below.ABC, BCD and EFG are triangles that each vertex is touching the circle.) <center> %02-14.png </center> -@02a.htm#2.5.3.4, click here to investigate by computer or GC/html (2.5.3.4)