Brianchons Theorem Circumscribed Circles and Tangent Point Questions

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1. Construct a triangle ?ABC and its inscribed circle. We can assume that the angles are such that A> B> C. Then construct a second circle that is inscribed in the part of the triangle where C is located. It should touch the first circle and two of the sides of the triangle.

2. Construct a triangle ?ABC where all angles are less than 90 degrees. Construct the three heights. The intersections of the heights with the sides form a new triangle, ?A1B1C1. Send out a beam from A1 towards B1 and reflect it at B1 (angle of incidence is = angle of incidence). Where does the outgoing beam intersect the triangle? Finally, reflect this as well. Conclusion?

3. Five points U1, U2, A, B, C determine a conic section. Choose five points in the plane, no three on a straight line and give them the designations U1, U2, A, B, C. These five points will be on the cone section. Draw the lines q = AB and p = AC and form the point F = U1B U2C. F is thus the intersection of lines U1B and U2C. A sixth point X on the conic section can now be formed as follows: Send out a beam from U1, it intersects the line p at Y1. Then draw the line Y1F and see where it intersects the line q. Call this new point Y2. Finally, pull the beam U2Y2. Our searched point is X = U1Y1 U2Y2. Now change the beam from U1 and see how X changes. Have the trace on function enabled so you see the trace of X. Was it an ellipse, a hyperbola or maybe a parabola? Move your five original points and see if you create other conic sections.

4. Solve problem 8 in the textbook with the help of GG. The figure you are going to draw is given by the following text: You want to measure the distance between two points A and B in the terrain. However, it is impossible to go straight between the two points. Instead, first follow a path from A that forms 50 degrees with AB until you reach a point P. AP = 150 meters. At P, turn 20 degrees towards B along another path until you reach another point Q. P Q = 50. From Q you then turn 40 degrees on a path that leads directly to B. All paths are straight lines. Determine AB.

5. Three circles C1, C2 and C3 all touch each other (all with all) and each touches 2 on each of the following pages in a circumscribed square. Draw such a figure.

6. Place two points on the y-axis. A at (0.1) and B higher up. Now find 2 circles C1 and C2 with midpoints on the x-axis such that C1 should go through A and C2 through B. In addition, the keys in both cases should form 45 degrees with the y-axis. The midpoint of C1 shall have a positive x-coordinate and the midpoint of C2 shall have a negative x-coordinate. C1 and C2 intersect at a point C in the first quadrant. Calculate the angle, ?, between the keys at the point of intersection. You have now constructed a so-called hyperbolic triangle with angles 45 degrees, 45 degrees and ?. Now move point B so that ? comes as close to 45 degrees as possible.

7. Three circles, C1, C2, C3, are disjoint, which means that they are separated (no common points) and they have radii 1,2 and 3. Find the inscribed circle, C, which is tangent to all three circles (all three the circles are thus outside C). Here you need to use the hyperbolic tool. When two circles touch each other, a straight line runs through the midpoints and the tangent point. For a hyperbola, for each point on the hyperbola, the difference in distance to two fixed points is always the same.

8. Two triangles ABC and A1B1C1 that are perspectiveistic from a point O are also perspectiveistic from a line. Illustrate this with GeoGebra. Assume a triangle ABC and a point O outside it. Drag the line OA and select A1 on this line. Then draw the line OB and select B1 on this line and finally the line OC and on the C1. Select all three points on the same page about ?ABC. Then draw the lines AB and A1B1 they intersect at a point D. The lines AC and A1C1 they intersect at a point E and the lines BC and B1C1 they intersect at a point F. Illustrate with GeoGebra that D, E and F lie on a line.

9. Select a line l and two points F1 and F2 on this and a third point A outside the line. Draw an ellipse with F1 and F2 as focal points passing through point A. Mirror a line passing through one focal point and A and see that it is reflected in the other focal point. For reflection, you need the key to the ellipse at point A. Then illustrate with the help of GG that the sum of the distances from a point on the edge to the two focal points is constant. You must therefore move around the point and see that the sum does not change. Finally, take a point on the ellipse and send out a beam between the focal points F1 and F2. Do three reflections as above. Move your points and see if you can find a periodic course (think of a billiard ball) with three bounces against the ellipse. The track should therefore look like a V.

10. A tool for dividing an angle is usually called a tomahawk. Take a semicircle with radius r and diameter DB and center O. Extend the diameter one distance r to a point A so that | BA | You also need a normal to DA through B. ?UV W must be divided into three equal parts. Place V on the normal (on the same side as the semicircle). V U should pass through point A and V W should touch the semicircle. Check with GG that the sections V B and V O divide the given angle into three parts. Move around V on the normal and see that it always applies. If you add a similar tomahawk with the center in A that touches the first tomahawk in B, you can divide an angle into five equal parts. Try!

11. A point P is reflected in the three sides of a triangle. Illustrate in GG that the three mirror points lie on the same straight line then and only when P lies on the circle described for the triangle.

12. Construct (almost) a regular hexagon, heptagon (see the leaf on Moodle). You can draw circles with a radius that you specify.

13. Construct the following irregular pentagon: The distance a = 1, b = d = e = 1/2 and c = ? 1 2 (? 3?1), see figure. It has been found that with the help of such pentagons one can cover the whole planet. If you google Tesselations of the Plane with Pentagons, you will find your figure. Take a few steps in this mosaic. You can mirror points in lines and move points in parallel in a certain direction. You can also determine the color of the pentagons.

14. In a circle with radius r, a circular arc with a center angle of 90 degrees is dashed in the figure below. It is folded over (downwards) and this arc of a circle is solid in the figure. Then a smaller circle is inscribed symmetrically in the figure. Two equal claw-like areas are then formed. Calculate the area of these.

15. Construct an arbitrarily selected triangle ?ABC where all angles are less than 120 degrees. Then construct an equilateral triangle outward from each side so that you get three new equilateral triangles ?A1BC, ?AB1C and ?ABC1. Show with GG that the lines AA1, BB1 and CC1 are competitive (ie intersect at one point). Call the intersection point P. Illustrate to your students that this point inside the triangle minimizes the sum of the distances to the corners. You can do this by entering another point Q inside the triangle and calculating QA + QB + QC in the input field.

16. Illustrate Ceva’s kit for your students. There is an input field that you must use. You can read about Ceva’s theorem on the internet.

17. Show the nine-point circle to your students. You can read about nine-point circle on the internet.

18. Each triangle ?ABC has three circumscribed circles. Each of these circles touches one of the sides of the triangle on the outside and the extensions of the other two sides. Construct these three circles. Let A1 be the tangent point on BC, B1 be the tangent point on AC and C1 be the tangent point on AB. Show in GG that lines AA1, BB1 and CC1 are competing (i.e. they intersect at a point).

19. For a given distance AB experiment describe the amount of all possible in-centers for triangles ?ABC. In-center is the center of the inscribed circle in the triangle. You get to have the trace on function on when you move around point C. Come up with a hypothesis and try to argue for it with the help of GG.

20. In a circle with radius r, a circular arc with a center angle of 90 degrees is dashed in the figure below. It is folded over (downwards) and this arc of a circle is solid in the figure. Then a smaller circle is inscribed symmetrically in the figure. Two equal claw-like areas are then formed. Calculate the area of these.