Chapter 6: Triangle Center Constructions
In this packet, you will learn to find the orthocenter, incenter, circumcenter, and centroid of a triangle using just a compass.[br][br]You will also explore midsegment properties in triangles.[br][br]There is also an extra credit option in the physical paper packet (on the last page: the nine-point circle).
[size=150][size=200][b][u]#1[/u]: How to Drop a Perpendicular from a Point to a Line[/b][/size][/size][b][i][br][br][/i][i]Use the example image on the front page of the packet to guide your construction.[/i][/b][br][br]To drop a perpendicular from a point to a line, you can do the following:[br][br]1) Make a circle or arc, centered at P, that intersects the line at two points. [br][br]2) Place points at the two intersections (use the "INTERSECT" tool).[br]Let's call those points "D" and "E."[br][br]3) Make a circle or arc centered at D. [br][br]4) Use the "COMPASS" tool to make a circle with the same radius as the circle centered at D, but center this circle at E.[br][br]5) Locate the point [u]below[/u] the line where the circles centered at D & at E intersect. [br]Place a point (using the "INTERSECT" tool) at this intersection.[br][br]6) Use the "RAY" tool to create a perpendicular ray from point P, through the point you just found.[br][br]This ray should be perpendicular to the given line (test this by moving point P around).
[color=#ff7700][size=150][size=200][b][u]#2[/u]: Use Altitudes to Find the Orthocenter[br][/b][/size][/size][b][i][br]Use the example image on the second page of the packet to guide your construction.[/i][/b][/color][br][br]In any triangle, if you drop a perpendicular from one vertex to the opposite side, the segment you construct is called an [b][color=#ff7700]altitude[/color][/b]. The altitudes from A and from B are shown in the sample diagram (p. 2 in the packet). The altitude from C is not shown.[br][br]1) Drop a perpendicular from B to the opposite side.[color=#b45f06][br][list][*][color=#b45f06][i]**To avoid a visually confusing diagram, I suggest lightening your circles so that they are barely visible before you move on.**[/i][/color][/*][/list][/color][br]2) Drop a perpendicular from A to the opposite side.[br][br]3) Drop a perpendicular from C to the opposite side.[br][br]4) If you have constructed each altitude correctly, then they should always intersect at a single point. [color=#b45f06][br][list][*][color=#b45f06]You can [b]test[/b] this by [b]moving[/b] points A, B, and C around. No matter what the triangle looks like, the altitudes should always intersect at one point. [/color][i]If they don't, reset and try again![/i][/*][/list][/color][br]5) Use the "TEXT" tool to [b]label[/b] this intersection point as the [u][b][color=#ff7700]orthocenter[/color][/b][/u].[br][color=#b45f06][br][i]("ortho" means "right" or "straight". For example, an orthodontist straightens teeth. Because this center is constructed from segments that intersect the sides of the triangle at right angles, the intersection point is called the orthocenter).[/i][/color]
[b][size=150][color=#ff7700]Move points A, B, and C around to find different locations for the orthocenter.[/color][/size][/b][br][br]a) Try to find a triangle where the orthocenter is *outside* of the triangle.[br][br]b) Try to find a triangle where the orthocenter is *on* the edge of the triangle.[br][br]For what type of triangle is the orthocenter always [b][color=#ff7700]on[/color][/b] the triangle (not inside or outside)? [br]For what type of triangle is the orthocenter always [b][color=#ff7700]outside[/color][/b] the triangle?
[color=#0000ff][size=150][size=200][b][u]#3[/u]: Use Angle Bisectors to Find the Incenter[br][/b][/size][/size][b][i][br]Use the example diagrams on the third page of the packet to guide your construction.[/i][/b][/color][br][br]The three [b][color=#0000ff]angle bisectors[/color][/b] of a triangle also meet at one point, called the [b][color=#0000ff]incenter[/color][/b].[br]The angle bisectors from A and from C are shown in the sample diagram (p. 3 in the packet). The angle bisector from B is not shown.[br][b][br][color=#0000ff][u]Find the incenter[/u]: [/color][/b][br][br]1) Construct the angle bisector of angle A ([i]follow the handwritten instructions in p. 3 of the packet[/i]).[br][br][color=#0b5394][i]**To avoid a visually confusing diagram, I suggest lightening your circles so that they are barely visible before you move on.**[/i][br][/color][br]2) Construct the angle bisector of angle C.[br][br]3) Construct the angle bisector of angle B.[br][br]4) If you have constructed each angle bisector correctly, then they should always intersect at a single point. [br][list][*][color=#0b5394]You can [b]test[/b] this by [b]moving[/b] points A, B, and C around. No matter what the triangle looks like, the angle bisectors should always intersect at one point. [/color][i]If they don't, reset and try again![/i][/*][/list][br]5) Use the "TEXT" tool to [b]label[/b] this intersection point as the [u][b][color=#0000ff]incenter[/color][/b][/u].[br][br][b][color=#0000ff][u]Construct the inscribed circle[/u]:[/color][/b][br][br]6) Drop a perpendicular from the incenter to one side. [br][br]7) Place a point where it intersects the side of the triangle ([i]see the 2nd diagram on p. 3 of the packet[/i]).[br][br]8) Construct a circle centered at the incenter that goes through this point. [br][br]This circle should just [b][i]touch[/i][/b] all three sides of the triangle. We call this the[color=#0000ff] [b]inscribed circle[/b][/color] of the triangle.[br][list][*][color=#0b5394][b]Test[/b] that your construction is correct by moving around points A, B, and C. The inscribed circle should always touch each side. [i]If it doesn't, reset and try again![/i][/color][/*][/list]
[size=150][size=200][b][color=#6aa84f][u]#4[/u]: Use Perpendicular Bisectors to Find the Circumcenter[br][/color][/b][/size][/size][b][i][color=#6aa84f][br]Use the example image on the fourth page of the packet to guide your construction.[/color][br][/i][/b][br]The three [color=#6aa84f][b]perpendicular bisectors[/b] [/color]of a triangle also meet at one point, called the [b][color=#6aa84f]circumcenter[/color][/b].[br]The perpendicular bisector between B & C is shown in the sample diagram (p. 4 in the packet). The other perpendicular bisectors are not shown.[br][b][br][color=#6aa84f][u]Find the circumcenter[/u]: [/color][/b][br][br]1) Construct the perpendicular bisector between B & C.[br][list][*](See example on p. 4: Create circles with the [u]same[/u] radius, one centered at B and one centered at C. Use the "INTERSECTION" tool to place points at the two places where these circles intersect. Use the "LINE" tool to construct the perpendicular bisector through these points).[br][br][/*][/list][i]**To avoid a visually confusing diagram, I suggest lightening your circles so that they are barely visible before you move on.**[br][/i][br]2) Construct the perpendicular bisector between A & C.[br][br]3) Construct the perpendicular bisector between A & B. [br][br]4) If you have constructed each line correctly, then they should always intersect at a single point. [br][list][*][color=#274e13]You can [b]test[/b] this by [b]moving[/b] points A, B, and C around. No matter what the triangle looks like, the perpendicular bisectors should always intersect at one point. [/color][i][color=#274e13]If they don't, reset and try again![/color][/i][/*][/list][br]5) Use the "TEXT" tool to [b][color=#6aa84f]label[/color][/b] this intersection point as the [u][color=#6aa84f][b]circumcenter[/b][/color][/u].[br][br][b][u][br][color=#6aa84f]Construct the circumscribed circle[/color][/u][color=#6aa84f]:[/color][/b][br][br]6) Construct a circle centered at the circumcenter that goes through each [b]vertex[/b] (you can use the circumcenter's distance to A, B, or C to set the radius).[list][*][color=#274e13][b]Test[/b] that your construction is correct by moving around points A, B, and C. The circumscribed circle should always go through each vertex. [i]If it doesn't, reset and try again![/i][/color][/*][/list][br]The circumcenter is the center of the [b][color=#6aa84f]circumscribed circle[/color][/b], which is the circle that goes through each [b]vertex[/b] of the triangle.[br][i][br]("circum" means "around," like in the words circumference [distance around a circle] or circumnavigate [to go around the world]).[/i]
[b][size=150][color=#6aa84f]Move points A, B, and C around to find different locations for the circumcenter.[/color][/size][/b][br][br]a) Try to find a triangle where the circumcenter is *outside* of the triangle.[br][br]b) Try to find a triangle where the circumcenter is *on* the edge of the triangle.[br][br]For what type of triangle is the circumcenter always [b][color=#6aa84f]on[/color][/b] the triangle (not inside or outside)? [br]For what type of triangle is the circumcenter always [b][color=#6aa84f]outside[/color][/b] the triangle?
[color=#9900ff][size=150][size=200][b][u]#5[/u]: Use Medians to Find the Centroid[br][/b][/size][/size][b][i][br]Use the example image on the fifth page of the packet to guide your construction.[/i][/b][/color][br]A [b]median[/b] is a segment that connects a [b]vertex[/b] to the [b]midpoint [/b]of the opposite side.[br][br]The three [b][color=#9900ff]medians[/color][/b] of a triangle also meet at one point, called the [b][color=#9900ff]centroid[/color][/b].[br][br]The median from A to the midpoint between B & C is shown in the sample diagram (p. 5 in the packet). The other medians are not shown.[br][br]1) Construct the median from A to the midpoint between B & C.[br][list][*][i][b]To find the midpoint of a side:[/b][/i][/*][*] (1) construct the perpendicular bisector, just like in #4. [/*][*] (2) Place a point where the perpendicular bisector intersects the side. ([/*][*] (3) Hide the perpendicular bisector (use the color settings in the top right to bring the thickness of the line to "0").[color=#9900ff][br][/color][/*][/list][br]2) Construct the median from B to the midpoint between A & C.[br][br]3) Construct the median from C to the midpoint between A & B.[br][br]4) If you have constructed each median correctly, then they should always intersect at a single point. [br][list][*][color=#674ea7]You can [b]test[/b] this by [b]moving[/b] points A, B, and C around. No matter what the triangle looks like, the medians should always intersect at one point. [i]If they don't, reset and try again![/i][/color][/*][/list]5) Use the "TEXT" tool to [b]label[/b] this intersection point as the [u][b][color=#9900ff]centroid[/color][/b][/u][color=#9900ff].[/color]
[size=150][color=#9900ff][b]Examine the relationship between the centroid & each median:[/b][/color][/size][br][br]The centroid splits each median into two parts, one short and one long. How does the long part compare to the short part?[br][br]To answer the question, choose any one of the medians, and use the "COMPASS" tool to make a circle with the radius length = the short part. [br][br][b]Exactly how many short parts will fit into the long part?[br][br][/b][i]Check that this is always true by moving around A, B, and C, and checking (visually) that this relationship looks consistent.[/i]
[b][color=#9900ff]Extra Credit (in addition to the last page of the packet): [/color][br][br][/b]The centroid is the point that balances the area/weight of the object. In other words, a triangle will stay balanced when placed on its centroid.[br][br]Obtain a piece of cardboard, draw any triangle on it, locate its centroid, and cut out the triangle. It should balance at its centroid on the tip of the pin! To get this credit, share a photo showing how you found the centroid (the medians drawn on the triangle), and share a video of you balancing the triangle on the pin.
[size=150][size=200][b][color=#ff00ff][u]#6[/u]: Construct a Midsegment and Discern its Properties[br][/color][/b][/size][/size][b][i][color=#ff00ff][br]Use the example image on the sixth page of the packet to guide your construction.[/color][br][/i][/b][br]In #5, you saw how to find the midpoints of the sides of a triangle. In this lesson, you will connect two of those midpoints to form a midsegment.[br][br]1) Use the "POLYGON" tool to create [b]triangle ABC[/b].[br][br]2) Find the [b]midpoint between A & B[/b]. Find the [b]midpoint between A & C[/b] (using the steps to form a perpendicular bisector - see instructions in #5).[br][br]3) Use the "SEGMENT" tool to construct the [b]midsegment[/b].
[b][color=#ff00ff][size=150]Properties of Triangle Midsegments:[/size][/color][/b][br][br]Find [b]two[/b] interesting relationships between the midsegment and its opposite side.[br][br]Move points A, B, and C around to check that these relationships are always true.
[b][size=150][size=200][color=#ff0000]#7: The Nine-Point Circle (extra credit)[/color][/size][/size][/b][br][br]The extra credit page (the nine-point circle) is the last page in the physical packet. [br]I suggest trying this one by just making arcs with a physical compass, as it will look pretty complicated/messy when done using complete circles in Geogebra. [br]If you have questions or could use some help, let me know.