# The Midpoint Of The Points Of Tangency

Geometry Level 4

In right angled $$\triangle ABC$$, $$\angle ABC= 90°$$ and $$\angle BCA = 60°$$. The incircle of $$\triangle ABC$$ touches sides $$\overline{BC}$$ and $$\overline{AB}$$ at points $$D$$ and $$E$$ respectively. Let $$F$$ be the midpoint of $$DE$$. $$P, Q, R$$ are the feet of perpendiculars from $$F$$ on $$BC, CA, AB$$ respectively. Let $$k= \dfrac{AB+BC+CA}{PQ+QR+RP}$$. Find $$\left \lfloor 100k \right \rfloor$$.

Details and assumptions

• $$\lfloor x \rfloor$$ denotes the greatest integer function, i.e. it is the greatest integer not exceeding $$x$$. For example, $$\lfloor 2.3 \rfloor = 2, \lfloor \pi \rfloor = 3, \lfloor 5 \rfloor= 5.$$

• Once you find the exact form of $$k$$, you might use a calculator to proceed.

• The last digit of the answer that is being accepted is $$8$$. If your answer slightly differs from the intended one because of approximation errors, you should enter the closest integer whose last digit is $$8$$.

• GeoGebra users will be prosecuted.

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