UPSC CDS Math Circle Questions

Note: The option should be circumcentre instead of circumferenceLet OP be the pole inside the triangular park ABC.
Here ∠OAP = ∠OBP = ∠OCP = θ (Say)
In ΔAOP,
tan‌θ=

OP

OA

⇒OA=

OP

tan‌θ

...(1)
In right ΔBOP,
tan‌θ=

OP

OB

⇒OB=

OP

tan‌θ

....(2)
In right ΔCOP,
tan‌θ=

OP

OC

⇒OC=

OP

tan‌θ

...(3)
From (1), (2) and (3), we have
OA = OB = OC
So, O is equidistant from the vertices of the ΔABC. Thus, O is the circumcentre of ΔABC.
Therefore, the foot of the pole is at the circumcentre.