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Problems in GeometryPrithwijit De
ICFAI Business School, Kolkata
Republic of India
email: de.prithwijit@gmail.com
Problem 1 [BMOTC]
Prove that the medians from the vertices A and B of triangle ABC are
mutually perpendicular if and only if |BC|2 + |AC|2 = 5|AB|2 .
Problem 2 [BMOTC]
Suppose that ∠A is the smallest of the three angles of triangle ABC. Let D
be a point on the arc BC of the circumcircle of ABC which does not contain
A. Let the perpendicular bisectors of AB, AC intersect AD at M and N
respectively. Let BM and CN meet at T . Prove that BT + CT ≤ 2R where
R is the circumradius of triangle ABC.
Problem 3 [BMOTC]
Let triangle ABC have side lengths a, b and c as usual. Points P and Q
lie inside this triangle and have the properties that ∠BP C = ∠CP A =
∠AP B = 120◦ and ∠BQC = 60◦ + ∠A, ∠CQA = 60◦ + ∠B, ∠AQB =
60◦ + ∠C. Prove that
(|AP | + |BP | + |CP |)3 .|AQ|.|BQ|.|CQ| = (abc)2 .
Problem 4 [BMOTC]
The points M and N are the points of tangency of the incircle of the isosceles
triangle ABC which are on the sides AC and BC. The sides of equal length
are AC and BC. A tangent line t is drawn to the minor arc M N . Suppose
that t intersects AC and BC at Q and P respectively. Suppose that the lines
AP and BQ meet at T .
(a) Prove that T lies on the line segment M N .
(b) Prove that the sum of the areas of triangles AT Q and BT P is
minimized when t is parallel to AB.
Problem 5 [BMOTC]
In a hexagon with equal angles, the lengths of four consecutive edges are 5,
3, 6 and 7 (in that order). Find the lengths of the remaining two edges.
1
,Problem 6 [BMOTC]
The incircle γ of triangle ABC touches the side AB at T . Let D be the point
on γ diametrically opposite to T , and let S be the intersection of the line
through C and D with the side AB. Show that |AT | = |SB|.
Problem 7 [BMOTC]
Let S and r be the area and the inradius of the triangle ABC. Let rA denote
the radius of the circle touching the incircle, AB and AC. Define rB and
rC similarly. The common tangent of the circles with radii r and rA cuts a
little triangle from ABC with area SA . Quantities SB and SC are defined in
a similar fashion. Prove that
SA SB SC S
rA
+ rB
+ rC
= r
Problem 8 [BMOTC]
Triangle ABC in the plane Π is said to be good if it has the following property:
for any point D in space, out of the plane Π, it is possible to construct a
triangle with sides of lengths |AD|, |BD| and |CD|. Find all good triangles.
Problem 9 [BMO]
Circle γ lies inside circle θ and touches it at A. From a point P (distinct
from A) on θ, chords P Q and P R of θ are drawn touching γ at X and Y
respectively. Show that ∠QAR = 2∠XAY .
Problem 10 [BMO]
AP , AQ, AR, AS are chords of a given circle with the property that
∠P AQ = ∠QAR = ∠RAS.
Prove that
AR(AP + AR) = AQ(AQ + AS).
Problem 11 [BMO]
The points Q, R lie on the circle γ, and P is a point such that P Q, P R are
0
tangents to γ. A is a point on the extension of P Q and γ is the circumcircle
0
of triangle P AR. The circle γ cuts γ again at B and AR cuts γ at the point
C. Prove that ∠P AR = ∠ABC.
2
,Problem 12 [BMO]
In the acute-angled triangle ABC, CF is an altitude, with F on AB and BM
is a median with M on CA. Given that BM = CF and ∠M BC = ∠F CA,
prove that the triangle ABC is equilateral.
Problem 13 [BMO]
A triangle ABC has ∠BAC > ∠BCA. A line AP is drawn so that ∠P AC =
∠BCA where P is inside the triangle. A point Q outside the triangle is
constructed so that P Q is parallel to AB, and BQ is parallel to AC. R is the
point on BC (separated from Q by the line AP ) such that ∠P RQ = ∠BCA.
Prove that the circumcircle of ABC touches the circumcircle of P QR.
Problem 14 [BMO]
ABP is an isosceles triangle with AB=AP and ∠P AB acute. P C is the
line through P perpendicular to BP and C is a point on this line on the
same side of BP as A. (You may assume that C is not on the line AB). D
completes the parallelogram ABCD. P C meets DA at M . Prove that M is
the midpoint of DA.
Problem 15 [BMO]
In triangle ABC, D is the midpoint of AB and E is the point of trisection
of BC nearer to C. Given that ∠ADC = ∠BAE find ∠BAC.
Problem 16 [BMO]
ABCD is a rectangle, P is the midpoint of AB and Q is the point on P D
such that CQ is perpendicular to P D. Prove that BQC is isosceles.
Problem 17 [BMO]
Let ABC be an equilateral triangle and D an internal point of the side BC.
A circle, tangent to BC at D, cuts AB internally at M and N and AC
internally at P and Q. Show that BD + AM + AN = CD + AP + AQ.
Problem 18 [BMO]
Let ABC be an acute-angled triangle, and let D, E be the feet of the per-
pendiculars from A, B to BC and CA respectively. Let P be the point where
the line AD meets the semicircle constructed outwardly on BC and Q be the
point where the line BE meets the semicircle constructed outwardly on AC.
Prove that CP = CQ.
3
, Problem 19 [BMO]
Two intersecting circles C1 and C2 have a common tangent which touches
C1 at P and C2 at Q. The two circles intersect at M and N , where N is
closer to P Q than M is. Prove that the triangles M N P and M N Q have
equal areas.
Problem 20 [BMO]
Two intersecting circles C1 and C2 have a common tangent which touches C1
at P and C2 at Q. The two circles intersect at M and N , where N is closer
to P Q than M is. The line P N meets the circle C2 again at R. Prove that
M Q bisects ∠P M R.
Problem 21 [BMO]
Triangle ABC has a right angle at A. Among all points P on the perimeter
of the triangle, find the position of P such that AP + BP + CP is minimized.
Problem 22 [BMO]
Let ABCDEF be a hexagon (which may not be regular), which circumscribes
a circle S. (That is, S is tangent to each of the six sides of the hexagon.)
The circle S touches AB, CD, EF at their midpoints P , Q, R respectively.
Let X, Y , Z be the points of contact of S with BC, DE, F A respectively.
Prove that P Y , QZ, RX are concurrent.
Problem 23 [BMO]
The quadrilateral ABCD is inscribed in a circle. The diagonals AC, BD
meet at Q. The sides DA, extended beyond A, and CB, extended beyond
B, meet at P . Given that CD = CP = DQ, prove that ∠CAD = 60◦ .
Problem 24 [BMO]
The sides a, b, c and u, v, w of two triangles ABC and U V W are related by
the equations
u(v + w − u) = a2
v(w + u − v) = b2
w(u + v − w) = c2
Prove that triangle ABC is acute-angled and express the angles U , V , W in
terms of A, B, C.
4
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