The Art and Craft of Problem Solving

Problems and Exercises

8.3 SURVIVAL GEOMETRY II 279

Here is a wide selection of elementary problems and exercises. No geometric facts other than those
developed in this and the previous section are needed for their solution. Of course, "elementary"
does not mean "easy." However, most of the problems below should succumb to patient investiga­
tion, judicious choice of auxiliary objects, persistant angle chasing, location of similar triangles,
etc. You should attempt every problem below, or at the very least, read each problem, since many
of the ideas developed in the problems will be used later.

8.3.14 Let line e intersect three parallel lines at A, B,

and C. Let line t intersect the same three lines at D, E,

and F, respectively. Show that AB / BC = DE / EF.

8.3.15 (New York State Mathematics League 1992)

Let ABC be a triangle with altitudes CD and AE, with

BD = 3 ,DA =5,BE = 2. FindEC.

8.3.16 Let ABC be a right triangle with right angle at

B. Construct a square on the hypotenuse of this trian­

gle (externally to the triangle). Let M be the midpoint

of this square. Show that M B bisects LABC.

8.3.17 Prove Euler's Inequality: The circumradius

of a triangle is at least twice the inradius.

8.3.18 A Tangent Version of the Inscribed Angle The­

orem. Consider the diagram illustrating circle termi­

nology on page 264. Prove that LABD = LACB.

8.3.19 Let P be an arbitrary point in the interior of an

equilateral triangle. Prove that the sum of the distances

from P to each of the three sides is equal to the altitude

of this triangle.

8.3.20 Carry out the proof in Example 8.3.1 on

page 271, but for the figure below:

F DE C

\CSJ A B

8.3.21 Recall that a trapezoid is a quadrilateral with

two parallel sides, called bases. The height of a trape­

zoid is the distance between the bases. Suppose a

trapezoid has base lengths a, b and height h.

(a) Find the area of the trapezoid in terms of a, b, h.

(b) Find the length of the line segment parallel to

the bases that passes through the intersection of

the diagonals, and is bounded by the two other

(non-base) sides.

8.3.22 Recall that a rhombus is a quadrilateral, all of

whose sides have equal length.

(a) Prove that rhombuses are parallelograms.

(b) Prove that the diagonals of a rhombus are per­

pendicular bisectors of each other.

(c) Prove that the area of a rhombus is equal to half

the product of the lengths of the diagonals.

8.3.23 (Hungary 1933) Let circles Ii, 12 be tangent to

one another at P. A line through P intersects Ii,12 at

AI, A 2 , respectively. Another line through P intersects

Ii,12 at B 1 ,B 2 , respectively. Prove that M 1 PB 1 '"

M 2 PB 2. (The circles may be tangent internally or

externally).

8.3.24 (Hungary 1936) Let P be a point inside triangle

ABC such that

[ABP] = [BCP] = [ACP].

Prove that P is the centroid of ABC.

8.3.25 Let r be the inradius of a triangle with sides

a, b, c, and let K and s denote, respectively, the area of

the triangle and the semi-perimeter (half the perime­

ter). Prove that K = rs.

8.3.26 Prove the Power of a Point theorem (8.1.1) for

the other two cases: P inside the circle, and P on the

circle.

8.3.27 The Distance from a Point to a Line. Let

ax + by + c = 0 be the equation of a line in the cartesian

plane, and let (u, v) be an arbitrary point in the plane.

Then the distance d from (u, v) to this line is given by

the well-known formula

d

_ lau + bv + cl

• v'
  a^2 +b^2

.

This can be proven with standard analytic geometry:

Find the coordinates of the foot of the perpendicular

from (u, v) onto the line by finding an equation for the

perpendicular (its slope will be the negative reciprocal