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Belarus — Selection and Training Session

Geometry Difficulty 5.5 AIME, harder Prove it Belarus

Point $L$ is marked on the side $AB$ of a triangle $ABC$ . The incircle of the triangle $ABC$ meets the segment $CL$ at points $P$ and $Q$ .
Is it possible that the equalities $CP = PQ = QL$ hold if $CL$ is

a) the median?
b) the bisector?
c) the altitude?
d) the segment joining vertex $C$ with the point $L$ of tangency of the excircle of the triangle $ABC$ with $AB$ ?

Solution

a) yes, it is possible; this is valid for the triangles with $AC = 5x$ , $AB = 10x$ , $BC = 13x$ , where $x$ is any positive real number;

b) it is impossible;

c) it is impossible;

d) yes, it is possible; this is valid for the triangles with $AC = 3x$ , $AB = 4x$ , $BC = 5x$ , where $x$ is any positive real number.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.