Geometry Mission Project

Name: ______

You will construct using only a compass and straight edge 5 Missions to demonstrate knowledge of the points of concurrency in Geometry. You may use different color pens or pencils to aid you in showing the points of concurrency. You must copy the triangle onto 5 sheets of 8 ½ by 11 sheets of “typing/printer paper” – that can be easily seen through (you will have to trace points from one mission to another).

1st: Copy the back of this worksheet 4 times on computer paper that can be seen through.

Mission #1 – The Circumcenter:

 Construct the Perpendicular Bisectors of the three sides: . Use the second construction in your earlier hand out to do this.

 Label the Midpoints: A,B, and C so A is the midpoint of , B is the midpoint for and C is the midpoint for

 The point where the bisectors meet is the Circumcenter. Label it P.

 Draw a circle with a center of P and a radius of

Mission #2 – The Centroid:

 Bisect the three sides: . Use the second construction in your earlier hand out to do this. The point where the bisector intersects are your midpoints

 Label the Midpoints: R, S, and T

 Draw the three Medians: Connect Vertices to Midpoints so you have the following Medians: FR, GT, and KS

 The point where the Medians meet is the Centroid. Label it M

Mission #3 – The Orthocenter:

 Construct the Altitudes to the three sides: Use the fourth construction in your earlier hand out to do this.

 Label the FEET of the altitudes: Q, N, and E

 The point where the altitudes meet is the Orthocenter. Label it O.

 Find the Midpoints of: Use the second construction in your earlier hand out to do this.

 Label the points: H, J, and Z.

Mission #4 – The Incenter:

 Bisect the three angles of the triangle. Use the last construction in your earlier hand out to do this.

 The point where the bisectors meet is the Incenter. Label it I.

 From the point I, construct a line that is perpendicular to the segment . Use the fourth construction in your earlier hand out to do this.

 Label the foot of the perpendicular W.

 With a radius of IW inscribe a circle within the triangle.

Mission #5 – Euler’s Nine – Point Circle:

 Use Mission #1 to copy the points A, B, C and P onto this page.

 From Mission #3, copy the points Q, N, E, H, J, Z, and O onto this page

 Draw line segment

 Bisect and label the Midpoint V.

 With the center of the compass at V and a radius equal to draw a circle.

 How many of the marked points are on the circle? Write this number on the Mission #5 page as a sentence.

 Note: Poncelot named this circle “The Nine – Point Circle.” French Geometers called it “Euler’s Circle.” German geometers refer to it as “Feuerbach’s Circle.”

Mission # ______

Name: ______

The Point of Concurrency demonstrated is: ______.

List all the information you know from this construction about the line segments and distances created: