Exam (elaborations)
Geometry Problem Set III
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Geometry Problem Set III
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Geometry Problem Set IIIMaths Olympiad Preparation
17th February 2024
1 Problems
1. Let two circles intersect at points X & Y. The length of their
common tangents is l and XY extended meet the two common
tangents at the points P & Q respectively. Prove that
P Q2 = XY 2 + l2 .
2. Let the angle bisector of ∠A of ∆ABC intersect the side BC at D.
Let P and Q be the points where AD meets the common tangents
of circles (ABD) & (ACD). Prove that
P Q2 = AB · AC.
3. ABCD is a square with incircle Γ. A tangent l to Γ meets the
sides AB and AD and the diagonal AC at P, Q, and R respectively.
Prove that
AP AR AQ
+ + = 1.
PB RC QD
4. Let D & E be points on the side BC of ∆ABC with inradius r such
that BD = BA & CE = CA. PA & QA are points of AC & BC re-
spectively such that DPA ∥ BC & EQA ∥ AC. Let NA = DPA ∩EQA .
Define the points PB , QB , PC , QC , NB & NC analogously. Prove
that
s s s
1 1 1 1 1 1 1
2 + 2 + 2 + 2 + 2 + 2 = r.
PA QA ANA PB QB BNB PC QC CNC
1
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5. In the previous problem, let la , lb & lc denote the lengths of the
angle bisectors from the vertices A, B & C respectively. Prove
that
la lb lc
+ + = 1.
la + ANA lb + BNB lc + CNC
6. Let M be a point on the side BC of ∆ABC such that the triangles
ABM & ACM have equal inradii. Prove that
AM 2 = s(s − a).
7. Let D be a point on the side BC of ∆ABC and let E & F be the
incentres of the triangles ABD & ACD respectively. If B, E, F & C
are concyclic, prove that
AD + DB AB
= .
AD + DC AC
8. In ∆ABC with incentre I, prove that for any point P, we have
a · AP 2 + b · BP 2 + c · CP 2 = abc + (a + b + c)IP 2 .
9. Prove that for any point M on the incircle of triangle ABC,
M A2 M B2 M C2
+ + = 2R + r.
ha hb hc
where ha , hb and hc are the lengths of the altitudes from A, B and
C respectively, while R and r denote circumradius and inradius,
respectively.
Happy Problem Solving!
2 Maths Olympiad Preparation
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