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Geometry Problem Set II
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Geometry Problem Set II
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Geometry Problem Set IIMaths Olympiad Preparation
16th February 2024
1 Problems
1. Let the incircle of an equilateral ∆ABC touch the sides BC, CA, AB
respectively at A′ , B ′ , C ′ . Let M be any point on the minor arc
B ′ C ′ , and H, K, L the orthogonal projections of M onto the sides
BC, CA, AB respectively. Prove that
√ √ √
M H = M K + M L.
2. Let I be the incentre of ∆ABC such that AI extended meets the
circumcircle at the point D. Let E, F be on BD, CD, respectively
such that IE, IF are perpendicular to BD, CD, respectively. If
IE + IF = AD
2 , find the value of ∠BAC.
3. Let ω be the circumcircle of the triangle ABC with ∠B = 3∠C.
The internal angle bisector of ∠A, intersects ω and BC at M and
D, respectively. Point E lies on the extension of the line M C
from M such that M E is equal to the radius of ω. Prove that
circumcircles of triangles ACE and BDM are tangent.
4. AB is a diameter of circle ω(O, R). From a point P on ω, perpen-
dicular P D is dropped on AB. The perpendicular bisector of P D
meets ω at the points X & Y respectively. Find all position of
point P such that
P X · P Y = P D2 .
Happy Problem Solving!
1
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