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Science
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I.F. Sharygin
Problems in Plane Geometry
Mir
Publishers
Moscow
Translated from Russian
by Leonid Levant
First published 1988
Revised from the 1986 Russian edition
m
17020'0000-304
21-88
2
056(01)-88
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Ha ane uucIWM

Theorems. Affine Prob-
lems
70. Loci of Points 82.
Triangles. A
Triangle
and a
Circle 89. Quadrilaterals tt7.
Circles aod
Tangents. Feuer-
bach's
Theorem t29. Combina-
tions
of Figures.
Displacements
in
the
Plane.
Polygons t37. Geo-
metrical Inequalities. Problems
on Extrema
147
Answers, Hint!, Solutions
too
Section t too
Section 2 . . . 214
Appendix. Inversion 392
Preface to the English Edition
This is a translation from the revised
edition of
the

lems have been added to
the
1982 edition,
the simpler problems in
the
first addition
having been eliminated, and a number of
new sections- (circles and tangents, poly-
gons, combinations of figures, etc.) having
been introduced, The general structure of
the
book has been changed somewhat to
accord
with
the new, more detailed, clas-
sification of the problems. As a result, all
the problems in
this
volume have been
rearranged.
Although the problems in this collection
vary in "age" (some of them
can be found
in old books and journals, others were
offered
at
mathematical olympiads or pub-
lished in the journal "Quant" (Moscow»,
I
still

collection
of geometrical puzzles aimed
at
demonstrating
the
elegance of elementary
geometrical techniques of proof and methods
of computation (without using vector alge-
bra
and
with
a minimal use of
the
method
of coordinates, geometrical transforms-
tions, though a somewhat wider use of trig-
onometry).
In
conclusion, I should like to
thank
A.Z. Bershtein who assisted me in prepar-
ing
the
first section of
the
book for print.
I am also grateful to A.A. Yagubiants who
let
me know several elegant geometrical
facts.

circumscribed
about
a triangle is equal
to
the
ratio
of
its
side to
the
sine of
the
opposite angle.
4.
Let
the vertex of an angle be located
outside a circle, and
let
the
sides of the
angle intersect
the
circle. Prove
that
the
angle is measured by
the
half-difference of
the arcs inside
the

to
the
circle at the point
A.
Prove
that
either of
the
two angles between
AB
and l is measured by
the
half-arc of the
Sec. f. Fundamental Facts
9
circle enclosed inside
the
angle
under
con-
sideration.
7.
Through
the
point
M located
at
a dis-
tancea from
the

equals
a
2
-
R2 (which is
the
squared
length
of
the
tangent).
8. A chord
AB
is
drawn
through
the
point
M
situated
at
a distance a from
the
centre
of a circle of
radius
R (a <
R).
Prove
that

AB
I I
AC
I.
The
same is
true
for
the
bisector of
the
exterior angle of
the
triangle.
(In
this
case
the
.point M lies on
the
extension of
the
side
Be.)
10.
Prove
that
the
sum
of

mel
drawn to
the
side a
can
be computed by
the
formula
m
a
=
~
V2lJ2+ 2c
2-a
2
•
12.
Given two triangles
having
one ver-
tex
A in common,
the
other
vertices being
situated
on two
straight
lines passing
10 Problems in Plane Geometry

is equal to half the product of
its
diagonals'
and
the
sine of the angle between them.
'-
15. Prove
the
validity
of the following
formulas for
the
area of a triangle:
a
l
sin B
sin
C
2
S =
2.
A'S
= 2R
SID
A
SID
B sm C,
sm
where

a
the
angle between them, and l
the bisector of this angle, then
(£
2abc08
T
1=
a+b
Sec. 1. Fundamental Facts
it
18. Prove
that
the
distances from
the
vertex A of
the
triangle
ABC
to
the
points
of tangency of
the
inscribed circle with
the
sides
AB
and

20. (a) Prove
that
the
altitudes
in a
triangle
'are
concurrent
(that
is intersect
at
one point). (b) Prove
that
the distance
from
any
vertex of a
triangle
to
the
point
of intersection of
the
altitudes
is twice
the
distance from
the
centre of the cir-
cumscribed circle to

other
side of
the
angle.
22. The hypotenuse of a
right
triangle
is
.equal to c, one of
the
acute angles being
30°.
Find
the
radius
of
the
circle with
centre
at
the
vertex of
the
angle of 30°
which separates
the
triangle into two
equi
v-
alent

to m and divides the
right
angle
in
the
ratio
1 : 2.
Find
the area of
the
triangle.
25. Given in a triangle
ABC
are three
sides:
I
Be
I = a, I CA I = b, I
AB
I = c.
Find
the
ratio
in which
the
point of inter-
section of
the
angle bisectors divides
the

sides is equal to
the
altitude
of
this
triangle.
28.
In
an isosceles triangle ABC, taken
on
the
base AC is a point M such
that
I
AM
I = a, I
Me
I = b. Circles are in-
scribed in
the
triangles
ABM
and
CBM.
Find
the
distance between
the
points
at

Sec. 1. Fundamental Facts
13
3t
. Determine the acute angle of the
rhombus whose side is the geometric
mean
of
its
diagonals.
32. The diagonals of a convex quadrilat-
eral are equal to
a and b, the line segments
joining the midpoints of the opposite sides
are congruent. Find
the area of the quadri-
lateral.
33. The side
AD
of the rectangle ABCD
is three times the side
AB;
points M and
N divide
AD
into three equal parts. Find
LAMB
+
LANB
+
LADB.

are chosen
80
that
the circle is divided
into three arcs in
the ratio 3 : 4 : 5. At the
division points, tangents are drawn to the
circle. Find the area of the triangle formed
by the tangents.
37. An equilateral trapezoid is circum-
scribed about a circle,
the
lateral side of
the trapezoid is
I, one of
its
bases is equal
to
4.
Find the area of the trapezoid.
38. Two straight lines parallel to the
bases of a trapezoid divide each lateral
14
Problems in Plane Geometry
side into three equal parts. The entire
trapezoid is separated by the lines into
three parts. Find the area of the middle
part
if the areas of the upper and lower
parts are 8

ratio
of
the
parallel sides is k. Find
the
angle
at
the
base.
42.
In
a trapezoid
ABCD,
the base
AB
is equal to a, and
the
base CD to b. .Find
the area of
the
trapezoid if the diagonals
of the trapezoid are known to be
the
bisec-
tors of the angles
DAB
and ABC.
43.
In
an equilateral trapezoid, the mid-

•
Find
the area of the trapezoid.
46. In a triangle
ABC,
the angle
ABC
is
tX. Find the angle AOe, where 0 is the
centre of the inscribed circle.
47. The bisector of the
right
angle is
drawn in a
right
triangle. Find the distance
between the points of intersection of the
altitudes of the triangles thus obtained,
if the legs of the given triangle are
a and b.
48. A straight line perpendicular to two
sides of a parallelogram divides the
latter
into two trapezoids in each of which a
circle can .be inscribed. Find the acute
angle of the parallelogram if its sides are
a and b (a < b). .
49. Given a half-disc with diameter
AB.
Two

16
Problems in Plane
Geometry
with
radius
a/y'2.
Find
the
area of
the
part
of
the
hexagon
not
enclosed
by
these
circles.
52. A
point
A is
taken
outslde a circle
of
radius
R. Two secants are drawn from
this
point: one passes through
the

distance between
the
centres
of two circles one of which passes through
the
points
D, A
and
B,
the
other
through
the
points B, C,
and
D.
54.
On
the
sides
AB
and
AD of
the
rhombus ABCD points M
and
N are
taken
such
that

I:
I
MN
I:
INB
I =
1:
2 : 3.',Through
the
points
M
and
N
straight
Iines are
drawn
parallel
to tHe side AC.
Find
the
area of
the
part
of
the
triangle
enclosed
between these lines if
the
area of

t.
Fundamental
Facts
i7
circle inscribed in
the
triangle
ABC
lies
on
the
given circle.
57. A circle is circumscribed about
an
equilateral
triangle ABC, and an
arbitrary
point
M is
taken
on
the
arc BC. Prove
that
I
AM
I = I
EM
I + I CM
I·

the
side
of
the
rhombus.
60. A square
with
side a is inscribed in
a circle.
Find
the
side of
the
square in-
scribed in one of
the
segments
thus
ob-
tained.
61. In a 120
0
segment of a circle
with
altitude
h a rectangle
ABCD
is inscribed
so
that

63. Express
the
side of a regular decagon
in terms of
the
radius
R of
the
circumscribed
circle.
64. Tangents
MA and
MB
are
drawn
from an exterior
point
M to a circle of
radius
R forming an angle a. Determine
2-01557
18
Problems in Plane Geometry
the
area of
the
figure bounded by the
tan-
gents and
the

circle pass-
ing through two neighbouring vertices of
the
rhombus and touching
the
opposite
side of
the
rhombus or
its
extension.
67. Given three pairwise tangent circles
of radius
r. Find
the
area of
the
triangle
formed by three lines each of which touches
two circles and does
not
intersect
the
third
one.
68. A circle of radius r touches a
straight
line
at
a

the
side CD a
point
N such
that
2 ICN I = I ND
I.
Find
the radius of
the
circle inscribed in
the
triangle
AMN.
70. Given a square
ABeD
with
side a.
Determine
the
distance between
the
mid-
point of
the
line segment
AM,
where M is
Sec. 1. Fundamental Facts
19

In
a
right
triangle
ABC
the leg CA
is equal to b, the leg
eB
is equal to a, CH
is
the
altitude,
and
AM
is the median.
Find
the
area of the triangle
BM
H.
73. Given an isosceles triangle
ABC
whose
LA
= a > 90° and I
Be
I = a. Find
the
distance between
the

a point A
taken
at
a distance
a from
the
centre. Find
the
radius of another
circle which is tangent to
the
diameter
at
the
point
A and touches
internally
the
given circle.
76.
In
a circle, three pairwise intersecting
chords are drawn. Each chord is divided
into
three equal
parts
by
the
points of
intersection. Find the radius of

triangle
ABC
whose
side is
equal
to a,
the
altitude
BK
is drawn.
A circle is inscribed
in
each of
the
triangles
ABK
and
BCK,
and
a common
external
tangent,
different from
the
side AC, is
drawn
to them.
Find
the
area of

Find
the
angle ACD.
80.
In
an inscribed
quadrilateral
ABeD
whose diagonals
intersect
at
a
point
K,
lAB
I = a,
18K
1=
b,
IAK
1=
c, I
CDI=
d.
Find
I
AC
I.
8t. A circle is circumscribed
about

ratio
of
the
area of
the
circle to
the
area of
the
trapezoid.
82.
In
an
equilateral
trapezoid
ABCD,
the base
AD
is
equal
to a,
the
base
Be
is
equal
to b, I
AB
I = d.
Drawn

squares of
the
Sec. 1. Fundamental Facts
21
distances
from
the
point
M
taken
on a
diam-
eter
of a
circle
to
the
end
points
of any
chord
parallel
to
this
diameter
if
the
radius
of
the

the
circles if
the
distance
between
their
centres
is
equal
to a.
85.
Given
a
regular
triangle
ABC. A
point
K divides
the
side
A C in
the
ratio
2 : 1,
and
a
point
M
divides
the

about
the
triangle
ABC.
86. Two circles of
radii
Rand
R/2
touch
each
other
externally.
One of
the
end
points
of
the
line
segment
of
length
2R forming
an angle of
30°
with
the
centre
line
coincides

and
an
altitude
AD
are
drawn
in a
triangle
ABC.
Find
the
side
AC
if
it
is known
that
thelines
EX
and
BE
divide
the
line
segment
AD
into
three
equal
parts

inscribed in the triangle.
90. Find the area of the pentagon bounded
by the lines
BC, CD,
AN,
AM,
and
BD,
where
A,
B, and D are the vertices of a
square
ABCD,
N the midpoint of the side
BC,
and M divides the side CD in the
ratio
2 : 1 (counting from the vertex C) if the
side of the square
ABCD
is equal to a.
91. Given in a triangle
ABC:
LBAC
=
a,
LABC
=~.
A circle centred
at

BN
I = I
NC
I). K lies on
the
side
DA
(2 I
DK
I = I
KA
I). Find the sine
of
the
angle between the lines MC and N K.
94. A circle of radius r passes through the
vertices A and B of the triangle
ABC
and