Construction Pod Game: Part E

Can you create a triangle with the polygon tool and construct its centroid? If you construct an isosceles triangle, where is its centroid? How about for a right triangle?

If you construct a circle with its center at the circumcenter of any triangle and its radius going to one of the triangle's vertices, the circle will go through all three vertices. That is the definition of the "circumcenter" (the center of the circumference or circle of the triangle). Were you able to construct the circumcenter of your own triangle? Did you drag the vertices to see if the circumcenter is always inside the triangle? Do you wonder why all three perpendicular bisectors of the sides meet at the same point? (Remember that a point is defined by just two lines crosing.)

The "altitude" of a triangle is the line segment from the base of the triangle perpendicularly to the opposite vertex. If you take AB as the base, then FC is the altitude, if FC is perpendicular to AB. You may know that the area of a triangle is 1/2 * base * altitude. How would you prove this? Construct a rectangle and connect two opposite vertices with a diagonal line segment, forming two congruent right triangles. The area of the rectangle is the base * height. So what is the area of each right triangle? This proves a special case of a right triangle's area.

A circle with center at the incenter of a triangle and radius to a point where a vertex bisector meets a triangle side will be inscribed in the triangle. The inscribed circle will touch each side of the triangle at exactly one point (it will be "tangent" to the side). Can you construct a triangle with a circle inscribing the triangle and a circle inscribed inside the triangle? When do the two circles have the same center?

You can create new tools in GeoGebra. For instance, you can go back to your constructions of the centroid, circumcenter, orthocenter and incenter and make your own custom tools. Then you can use your custom tools to place each of these points in a new triangle here. To define a custom tool, go to the GeoGebra menu under Tools and select Create New Tool. Follow the steps: 1. select the triangle and the special point as output objects, 2. select the triangle vertices as input objects, 3. name the tool something like "centroid" and check "show in toolbar." You can also save your custom tools on your computer from the Manage Tools option under the Tools menu. Custom tools are powerful. They are shortcuts to doing complicated things and you know exactly how they work. You can develop your own mini-domains of geometry with them. You can add new functions, like copying angles and inscribing triangles in circles. When you drag your triangle with these four special points, do you notice any possible dependencies among them?

Describe the nine points on the circle. As you drag the vertices, do the nine points stay on the circle and do the circumcenter, incenter and orthocenter stay on the Euler segment, whose midpoint stays in the center of the 9-point circle? Here are many points and lines with complicated dependencies among themselves and the vertices of the triangle. Can you prove why the nine points are all on the same circle? Can you prove why the circumcenter, incenter and orthocenter are all on the same line segment, whose midpoint is the center of the circle. If you looked carefully at the detailed steps in constructing all these points, lines and circles, you could work out much of the proof -- often using equalities of congruent triangles proven by theorems like SSS, SAS and ASA.

Given the locations of the three trees, how would you construct the locations of the three pots of coins?

How did you construct the center of the circle? How did you figure out the radius length?

What additional lines did you have to construct to determine locations for points E and F?

Describe the different ways that you constructed triangles that are always congruent to triangle ABC no matter how you drag A, B or C.

Answer questions 1 through 7 in Challenge 47 in your own words.

Just do it! Invent a challenge for your team mates and others who have completed the Construction Pod Game. Why did you choose this topic?