Problems and Exercises8.2 SURVIVAL GEOMETRY I 269The problems below are elementary, in that no geometric facts other than those developed in this
section are needed for their solution. Of course, "elementary" does not mean "easy." Some of them
may be easier to do after you read the next section. But do try them all. The problems include
many formulas and important ideas that will be used in later sections.
8.2.24 Let ABC be isosceles with vertex angle A, and I. Centroid
let E and D be points on AC and AB, respectively, so 2. Circumcenter
that
AE =ED =DC =CB.
Find LA.
8.2.25 In the previous problem, we chose points on
the equal sides of an isosceles triangle to create a
"chain" of four equal segments, including the base of
the triangle. Can you generalize to n-segment chains?
8.2.26 Prove that the midpoints of the sides of an arbi
trary quadrilateral are the vertices of a parallelogram.
8.2.27 Let ABC be a right triangle, with right angle
at C. Prove that the length of the median through C
is equal to half the length of the hypotenuse of ABC.
Thus if CE is the median, we have the nice fact that
AE =BE=CE.
8.2.28 Compass-and-Ruler Constructions. Undoubt
edly, you learned at least a few compass-and-ruler con
structions (also called Euclidean constructions). Re
view this important topic by finding the following con
structions (make sure that you can prove why they
work). You may use a compass and an unmarked ruler.
No other tools allowed (besides a pencil).
(a) Given a line segment, find its midpoint.
(b) Given a line segment, draw its perpendicular bi
sector.
(c) Given a line e and a point P not on e, draw a line
parallel to e that passes through P.
(d) Given a line e and a point P not on e, draw a line
perpendicular to e that passes through P.
(e) Given an angle, draw its bisector.
(f) Given a circle, find its center.
(g) Given a circle and a point exterior to it, draw the
tangent to the circle through the point.
(h) Given a line segment of length d, construct an
equilateral triangle whose sides have length d.
(i) Given a triangle, locate the- Orthocenter
- Incenter
U) Given a circle, with two given points P. Q in its
interior, inscribe a right angle in this circle, such
that one leg passes through P and one leg passes
through Q. The construction may not be possi
ble, depending on the placement of P, Q.
(k) Given two circles, draw the lines tangent to
them.
8.2.29 The triangle formed by joining the midpoints
of the sides of a given triangle is called the medial tri
angle.
(a) Prove that the medial triangle and the original
triangle have the same centroid. (For a much
harder variation on this, see Problem 8.3.41.)
(b) Prove that the orthocenter (intersection of alti
tudes) of the medial triangle is the circumcenter
of the original triangle.
8.2.30 Let e, and e 2 be parallel lines, and let OJ and
y be two circles lying between these lines so that PI is
tangent to OJ, OJ is tangent to y, and y is tangent to P 2.
Prove that the three points of tangency are collinear,
i.e., lie on the same line.
8.2.3 1 (Leningrad Mathematical Olympiad 1987) Al
titude CH and median BK are drawn in an acute trian
gle ABC, and it is known that BK = CH and LKBC =
LHCB. Prove that triangle ABC is equilateral.
8.2.32 (Leningrad Mathematical Olympiad 1988)
Acute triangle ABC with LBAC = 30 ° is given. Alti
tudes BB, and CC, are drawn; B 2 and C 2 are the mid
points of AC and AB, respectively. Prove that segments
B,C 2 and B 2 C, are perpendicular.
8.2.33 (Mathpath 2006 Qualifying Quiz) Consider the
following "recipe" for folding paper to get equilateral
triangles (see the figure below):