Sample Exam II

a) The incenter of a triangle is ...

the point of intersection of the internal angle bisectors of the triangle (or the center of the inscribed circle).

b) The Euler line of a triangle is the line segment determined by ...

the circumcenter (O) and the orthocenter (H) of the triangle.

c) The centroid of a triangle is ...

the point of intersection of the medians of the triangle.

d) A constructible number is the length of a line segment which ...

can be obtained from a unit length by straightedge and compass constructions.

e) The 9-point circle of a triangle contains the following 9 points of a triangle :

the three midpoints of the sides of the triangle, the three feet of the altitudes of the triangle and the three midpoints of the segments drawn from the vertices to the orthocenter of the triangle.

2. Prove that the internal bisectors of two angles of a triangle and the external bisector of the third angle intersect the opposite sides of the triangle in three collinear points.

Solution:

3. Using compass and straightedge construct the orthocenter of the triangle below.

The orthocenter is the point where the three altitudes of the triangle meet. By constructing two altitudes we can obtain the orthocenter as their point of intersection. To construct an altitude, we drop a perpendicular from a vertex to the opposite side. This is done as follows:

The orthocenter is thus constructed as follows:

4. In an arbitrary triangle ABC, let D be the foot of the altitude drawn from A to BC and let A', B' and C' be the midpoints of the sides of the triangle. Prove that DB'C'A' is an isosceles trapezoid.

Solution: Since B' and C' are midpoints of the sides of the triangle, the line joining them is parallel to the third side of the triangle, which includes the line segment DA'. Thus, the opposite sides, B'C' and DA' of this quadrilateral are parallel, so it is a trapezoid. Since, A' and C' are midpoints of the sides of the triangle, A'C' has length equal to 1/2 the length of AC. In the right triangle, ADC, the line segment DB' joins the right angle vertex to the midpoint of the hypotenuse. The length of such a line is 1/2 of the hypotenuse, which is AC. Thus, DB' and A'C' have the same length, so the trapezoid is isosceles.

5. Write a paragraph that describes the three famous construction problems of antiquity. Be sure to include in your paragraph a statement about why each of these problems has no solution.

Solution: The three famous problems of antiquity are the following:

• The Delian problem - duplicating the cube. The problem is to construct a cube that has twice the volume of a given cube. A particular instance of this problem would be to construct a cube whose volume is twice that of the unit cube. This entails constructing a side of the larger cube, and in this case that means constructing a length equal to the cube root of 2. This length is a root of the equation x³ - 2 = 0, but this cubic equation with rational coefficients has no rational root.
• Trisection of an Angle - The problem is to find the angle trisectors for an arbitrary angle. The general problem can not be done because it can't be done for some specific angles, for instance an angle of 60°. (Construction of 20 degree angle leads to the cubic equation 8x³ -6x - 1 = 0, and this does not have roots of the required type). (Wankel)
• Squaring the Circle - The problem is to construct a square that has the same area as the unit circle, i.e. . If this can be done, then the square root of would be constructible. And if that is true, then would also be constructible. But is a transcendental number (Lindemann, 1882), and such numbers are not constructible.