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Omni-MATH / An acute triangle ABC is inscribed in a circle of radius 1 with centre O; all the angles of ABC are greater than 45^(∘). B₁ is the foot of perpendicular from B to CO, B₂ is the foo…

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problem

An acute triangle ABCABC is inscribed in a circle of radius 1 with centre O;O; all the angles of ABCABC are greater than 45∘.45^\circ. B1B_{1} is the foot of perpendicular from BB to CO,CO, B2B_{2} is the foot of perpendicular from B1B_{1} to AC.AC. Similarly, C1C_{1} is the foot of perpendicular from CC to BO,BO, C2C_{2} is the foot of perpendicular from C1C_{1} to AB.AB. The lines B1B2B_{1}B_{2} and C1C2C_{1}C_{2} intersect at A3.A_{3}. The points B3B_{3} and C3C_{3} are defined in the same way. Find the circumradius of triangle A3B3C3.A_{3}B_{3}C_{3}.
Plain-text mathematical notation (without MathML)
An acute triangle ABC is inscribed in a circle of radius 1 with centre O; all the angles of ABC are greater than 45^(∘). 
B₁ is the foot of perpendicular from B to CO, B₂ is the foot of perpendicular from B₁ to AC. 
Similarly, C₁ is the foot of perpendicular from C to BO, C₂ is the foot of perpendicular from C₁ to AB. 
The lines B₁B₂ and C₁C₂ intersect at A₃. The points B₃ and C₃ are defined in the same way. 
Find the circumradius of triangle A₃B₃C₃.
Original LaTeX notation
An acute triangle $ABC$ is inscribed in a circle of radius 1 with centre $O;$ all the angles of $ABC$ are greater than $45^\circ.$ 
$B_{1}$ is the foot of perpendicular from $B$ to $CO,$ $B_{2}$ is the foot of perpendicular from $B_{1}$ to $AC.$ 
Similarly, $C_{1}$ is the foot of perpendicular from $C$ to $BO,$ $C_{2}$ is the foot of perpendicular from $C_{1}$ to $AB.$ 
The lines $B_{1}B_{2}$ and $C_{1}C_{2}$ intersect at $A_{3}.$ The points $B_{3}$ and $C_{3}$ are defined in the same way. 
Find the circumradius of triangle $A_{3}B_{3}C_{3}.$

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Source and history

task_id
9c841c4d-036f-5343-b92f-ff82bdf1dd0d
task_key
test--9c841c4d-036f-5343-b92f-ff82bdf1dd0d
task_revision_id
1
config / split
/ test

Official source

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