Competition problem — Geometry — Maths Olympiad Prep

Library / /24 of 50

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.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.