# Euler's line

 Here are three ways to associate a point to a triangle. The vertices are movable. The whole picture can be moved or zoomed. For each point, make three general observations about how it is related to the triangle. Without invoking the "Constructions," can you guess how they are constructed? (There may be several equivalent constructions, each deserving attention!) Once you see the constructions, can you verify your observations? Can you suggest a relationship between the three points? This worksheet is a small modification of a worksheet of the same name created by Judah Schwartz: [url]https://tube.geogebra.org/user/profile/id/25758[/url]

# Point in an Equilateral Triangle

 The GOLD point is chosen at random inside the equilateral triangle. The red, blue and green segments are lines drawn from the GOLD point and perpendicular to each of the sides. Their lengths vary in size as you move the GOLD point from place to place inside the triangle. However, the sum of their lengths is constant. What is the sum equal to? Why? Can you prove it? What is happening when the GOLD point is outside the triangle? Would a similar thing be true in a square? Why or why not? What about other regular polygons with an odd number of sides? with an even number of sides? This worksheet is a modified version of a worksheet of the same name created by Judah Schwartz: [url]https://tube.geogebra.org/user/profile/id/25758[/url]

# Beyond the Median

 A standard result in geometry is that the medians of a triangle meet at a point and cut each other in the ratio of 2:1 Suppose we subdivide each leg of the triangle in two parts so that the ratio of the two parts is not 1:1 as in the case of medians, but rather n:1. Then we draw lines from the vertices to these points - let's call these lines n-dians. The three n-dians no longer intersect at a point - rather they define a triangle. What is the relationship between this triangle and the original triangle? What can you say about the lengths of the segments determined by the intersections of the n-dians? Can you prove {some, all} of your assertions? Note: You can use the spreadsheet to do calculations. For example ABC is the area of triangle ABC. Likewise GHI is the area of triangle GHI. This worksheet is a small modification of a worksheet of the same name created by Judah Schwartz: [url]https://tube.geogebra.org/user/profile/id/25758[/url]

# Ants Walking around a Triangle

 Two ants walk around a triangle. As indicated the point E and F are midpoints of their respective sides. The line DE is parallel to CB. The first ant follows the path B to C to A to B. The second and follows the path A to E to F to C to D to E to B. Which ant walks the longer distance?

# Quadrilateral with Inscribed Circle (Tangent Construction)

 This construction shows a quadrilateral with an inscribed circle. In this construction the points of tangency with the circle (T1, T2, T3, T4) can be moved to change the quadrilateral. What are some properties of such a quadrilateral? The spreadsheet shows some lengths. To show other lengths you can input them into the spreadsheet. The example of Distance[A,T1] is in the spreadsheet.

# Quadrilateral with Inscribed Circle (Vertex Construction)

 This construction shows a quadrilateral with an inscribed circle. In this construction the vertices A and C can be moved to change the quadrilateral. What are some properties of such a quadrilateral? The spreadsheet shows some lengths. To show other lengths you can input them into the spreadsheet. The example of Distance[A,T1] is in the spreadsheet.

# Yin Yang Yin

 Drag the white dots. How big is the green area? Does your formula continue to be correct when one or both of the dots are outside the original disk? This worksheet is a copy of a worksheet of the same name created by Judah Schwartz: [url]https://tube.geogebra.org/user/profile/id/25758[/url]

# Sum of Circle Areas

 The BLACK dot is connected to all four vertices of a rectangle. The length of the segment from each vertex to the BLACK dot is the radius of a circle - BLUE segments are the radii of BLUE circles; RED segments are the radii of RED circles. The BLACK dot and the vertices of the rectangle are moveable. Use the spreadsheet to explore relations involving the areas of the circles. What conjectures did you make? Can you prove them? Note: You can use the spreadsheet to do calculations, e.g. enter =B1 + B3 in a spreadsheet cell to get the sum Area B_1 + Area B_3 This worksheet is a small modification of a worksheet of the same name created by Judah Schwartz: [url]https://tube.geogebra.org/user/profile/id/25758[/url]

# Three Tangent Circles with Spreadsheet

 From Polya - Mathematical Discovery Three circles, each of which is tangent to the other two have their centers on a line. If the radius of the surrounding circle is R and the length of the vertical (black) segment is T what is the size of the green area? You can experiment with changing the sizes of the red and yellow circles by moving the WHITE dot, but it remains for you to prove your method of finding the size of the uncovered green area will work for any set of three mutually tangent circles like this. The spreadsheet shows the areas of the green, red and yellow circles: GG, RR, YY. It also shows R and T. You can use the spreadsheet to test your conjectures. (Note: the constant pi can be entered as 'pi' in the spreadsheet.) This worksheet is a small modification of a worksheet of the same name created by Judah Schwartz: [url]https://tube.geogebra.org/user/profile/id/25758[/url]

# Two Planets & A Sun - A Digital Triptych

 The motions of a Sun and two planets as seen in each of their coordinate systems. You can animate the motion by clicking on the l[b]ower left hand corner[/b] of the screen. You can show the trajectories by checking the trajectories checkbox. You can show the triangle formed by the the bodies by checking the triangles checkbox. Can you see a relationship between the apparent motion of Mars from Earth and the apparent motion of Earth from Mars? Can you explain the relationship you observe? Why does the triangle formed by the three bodies look the same in all three coordinate systems? An astronomical unit (AU) is approximately the distance between the Earth and the Sum. The initial setting has Mars at 1.5 AU from the sun, which is approximately correct. You can vary this setting and see how this effects the retrograde motion. (The speed of the planet depends on the distance from the sun. The worksheet computes the speed consistent with the laws of planetary motion.) What do you predict will happen as this distance increases? you This worksheet is a modified version of a worksheet of the same name created by Judah Schwartz: [url]https://tube.geogebra.org/user/profile/id/25758[/url]

# Burning tent on an elliptical island

 Our hero is camping on a elliptical island. From a distance she sees her tent burning. She runs toward the water with her GOLD pail in hand, fills it and heads for her tent. Toward what point on the shore should she run to travel the shortest distance? Does this strategy always work? Can the hero and tent be placed in spots so that getting water from any point on the shore would be equally good? You can modify the shape of the ellipse by moving the foci. Is the problem any simpler if you make the ellipse a circle? This worksheet is a modified version of a worksheet of the same name created by Judah Schwartz: [url]https://tube.geogebra.org/user/profile/id/25758[/url]

# An interesting Function on the space of quadrilaterals

 Drag the large BLACK dots {A, B, C, D} to form a quadrilateral you are interested in. This environment constructs a function from the space of quadrilaterals to the space of quadrilaterals as follows - --circumscribes a circle around points A,B,C with center at E --circumscribes a circle around points B,C,D with center at F --circumscribes a circle around points C,D,A with center at G --circumscribes a circle around points D,A,B with center at H and then constructs the quadrilateral EFGH (in GREEN) Explore this function and then write down all the conjectures you can about its behavior. Which of your conjectures can you prove? This worksheet is a small modification of a worksheet of the same name created by Judah Schwartz: [url]https://tube.geogebra.org/user/profile/id/25758[/url]

# The Three Roots of a Cubic Function

 This environment allows you to explore the following questions: Given three real numbers, how many cubic functions are there with those numbers as roots? How are these functions related to one another? Given a real number and a pair of complex conjugate numbers, How many cubic functions have these numbers as roots? How are these functions related to one another? Challenge – given two cubic functions with the same roots, can you find a formula relating them? What information do you need to compute do the computation? This worksheet is a small modification of a worksheet of the same name created by Judah Schwartz: [url]https://tube.geogebra.org/user/profile/id/25758[/url]

# Rational Functions & Asymptotic Behavior

 You can use this environment to explore the behavior of rational functions formed by dividing a constant, linear or quadratic function by a linear or quadratic function. Define the numerator function by dragging the red dots. Define the denominator function by dragging the blue dots. Fit the appropriate asymptote(s) to the rational function you have determined. Alternatively, choose a set of asymptotes and try to find a rational function with those asymptotes. This worksheet is a modified version of a worksheet of the same name created by Judah Schwartz: [url]https://tube.geogebra.org/user/profile/id/25758[/url]